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Edit detail for SandBox Quaternion Algebra is Frobenius in Many Ways revision 18 of 32

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Editor: Bill Page
Time: 2011/05/19 23:26:40 GMT-7
Note: bi-algebra conditions

changed:
-Cartan-Killing Trace Form
-\begin{axiom}
-
-( I I   ) / _
-(  Y Λ  ) / _
-(   Y I ) / _
-     V
-
-\end{axiom}
-
Multiplication is Associative
\begin{axiom}
test(
  ( I Y ) / _
  (  Y  ) = _
  ( Y I ) / _
  (  Y  ) )
\end{axiom}

added:
The Cartan-Killing Trace
\begin{axiom}

Ú:=
   (  Y Λ  ) / _
   (   Y I ) / _
        V
Ù:=
   (  Λ Y  ) / _
   ( I Y   ) / _
      V

test(Ù=Ú)

\end{axiom}
forms a non-degenerate associative scalar product for Y
\begin{axiom}
Ũ := r*Ù
test
     (    Y I    )  /
           Ũ        =
     (    I Y    )  /
           Ũ
determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ), j,1..dim), i,1..dim)
\end{axiom}

General Solution


changed:
-equate(f,g)==map((x,y)+->(x=y),ravel f, ravel g);
-eq1:=equate(d1,I);
-eq2:=equate(d2,I);
equate(eq)==map((x,y)+->(x=y),ravel lhs eq, ravel rhs eq);
eq1:=equate(d1=I);
eq2:=equate(d2=I);

added:

Co-associativity
\begin{axiom}
test(
  (  λ  ) / _
  ( I λ ) = _
  (  λ  ) / _
  ( λ I ) )
\end{axiom}

changed:
-Χ :=
H :=

changed:
-     (  I Y    )  =  Χ
     (  I Y    )  =  H

changed:
-     (    Y I  )  =  Χ
-
-\end{axiom}
-
-Bi-algebra
-\begin{axiom}
-)set output tex off
-)set output algebra on
-\end{axiom}
-\begin{axiom}
-bi1:=         _
     (    Y I  )  =  H

\end{axiom}

Bi-algebra conditions
\begin{axiom}
ΦΦ:=         _

changed:
-  (  Y Y  );
-
-bi2:=   _
-    Y / _ 
-    λ / _ 
-    Y / _
-    λ;
-
-test(bi1=bi2)
-\end{axiom}
-\begin{axiom}
-)set output algebra off
-)set output tex on
-\end{axiom}
  (  Y Y  ) ;

test( ΦΦ=H )
test( ΦΦ=H/H )
solve(equate(ΦΦ=H),Ξ(sb('p,[i]), i,1..#Ñ))
\end{axiom}

changed:
-H:𝐋 :=
Φ:𝐋 :=

changed:
-H0:𝐋 :=eval(H,ex1)$𝐋
-\end{axiom}
Φ0:𝐋 :=eval(Φ,ex1)$𝐋
\end{axiom}

Quaternion Algebra Is Frobenius In Many Ways

Linear operators over a 4-dimensional vector space representing quaternion algebra

Ref:

We need the Axiom LinearOperator? library.

axiom
)library CARTEN ARITY CMONAL CPROP CLOP CALEY
CartesianTensor is now explicitly exposed in frame initial CartesianTensor will be automatically loaded when needed from /var/zope2/var/LatexWiki/CARTEN.NRLIB/CARTEN Arity is now explicitly exposed in frame initial Arity will be automatically loaded when needed from /var/zope2/var/LatexWiki/ARITY.NRLIB/ARITY ClosedMonoidal is now explicitly exposed in frame initial ClosedMonoidal will be automatically loaded when needed from /var/zope2/var/LatexWiki/CMONAL.NRLIB/CMONAL ClosedProp is now explicitly exposed in frame initial ClosedProp will be automatically loaded when needed from /var/zope2/var/LatexWiki/CPROP.NRLIB/CPROP ClosedLinearOperator is now explicitly exposed in frame initial ClosedLinearOperator will be automatically loaded when needed from /var/zope2/var/LatexWiki/CLOP.NRLIB/CLOP CaleyDickson is now explicitly exposed in frame initial CaleyDickson will be automatically loaded when needed from /var/zope2/var/LatexWiki/CALEY.NRLIB/CALEY

Use the following macros for convenient notation

axiom
-- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])
Type: Void
axiom
-- list
macro Ξ(f,i,n)==[f for i in n]
Type: Void
axiom
-- subscript
macro sb == subscript
Type: Void

𝐋 is the domain of 4-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients.

axiom
dim:=4

\label{eq1}4(1)
Type: PositiveInteger?
axiom
macro ℒ == List
Type: Void
axiom
macro ℂ == CaleyDickson
Type: Void
axiom
macro ℚ == Expression Integer
Type: Void
axiom
𝐋 := ClosedLinearOperator(OVAR ['1,'i,'j,'k], ℚ)

\label{eq2}\hbox{\axiomType{ClosedLinearOperator}\ } (\hbox{\axiomType{OrderedVariableList}\ } ([ 1, i , j , k ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))(2)
Type: Type
axiom
𝐞:ℒ 𝐋      := basisOut()

\label{eq3}\left[{|_{\  1}}, \:{|_{\  i}}, \:{|_{\  j}}, \:{|_{\  k}}\right](3)
Type: List(ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer)))
axiom
𝐝:ℒ 𝐋      := basisIn()

\label{eq4}\left[{|^{\  1}}, \:{|^{\  i}}, \:{|^{\  j}}, \:{|^{\  k}}\right](4)
Type: List(ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer)))
axiom
I:𝐋:=[1]   -- identity for composition

\label{eq5}{|_{\  1}^{\  1}}+{|_{\  i}^{\  i}}+{|_{\  j}^{\  j}}+{|_{\  k}^{\  k}}(5)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
X:𝐋:=[2,1] -- twist

\label{eq6}\begin{array}{@{}l}
\displaystyle
{|_{\  1 \  1}^{\  1 \  1}}+{|_{\  i \  1}^{\  1 \  i}}+{|_{\  j \  1}^{\  1 \  j}}+{|_{\  k \  1}^{\  1 \  k}}+{|_{\  1 \  i}^{\  i \  1}}+ 
\
\
\displaystyle
{|_{\  i \  i}^{\  i \  i}}+{|_{\  j \  i}^{\  i \  j}}+{|_{\  k \  i}^{\  i \  k}}+{|_{\  1 \  j}^{\  j \  1}}+{|_{\  i \  j}^{\  j \  i}}+{|_{\  j \  j}^{\  j \  j}}+ 
\
\
\displaystyle
{|_{\  k \  j}^{\  j \  k}}+{|_{\  1 \  k}^{\  k \  1}}+{|_{\  i \  k}^{\  k \  i}}+{|_{\  j \  k}^{\  k \  j}}+{|_{\  k \  k}^{\  k \  k}}
(6)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
V:𝐋:=ev(1) -- evaluation

\label{eq7}{|^{\  1 \  1}}+{|^{\  i \  i}}+{|^{\  j \  j}}+{|^{\  k \  k}}(7)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
Λ:𝐋:=co(1) -- co-evaluation

\label{eq8}{|_{\  1 \  1}}+{|_{\  i \  i}}+{|_{\  j \  j}}+{|_{\  k \  k}}(8)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

Now generate structure constants for Quaternion Algebra

The basis consists of the real and imaginary units. We use quaternion multiplication to form the "multiplication table" as a matrix. Then the structure constants can be obtained by dividing each matrix entry by the list of basis vectors.

Split-complex and co-quaternions can be specified by Caley-Dickson parameters (q0 = -1, q1 = -1)

axiom
--q0:=sb('q,[0])
q0:=1  -- not split-complex

\label{eq9}1(9)
Type: PositiveInteger?
axiom
q1:=sb('q,[1])

\label{eq10}q_{1}(10)
Type: Symbol
axiom
--q1:=1  -- co-quaternion
QQ := ℂ(ℂ(ℚ,'i,q0),'j,q1);
Type: Type

Basis: Each B.i is a quaternion number

axiom
B:ℒ QQ := map(x +-> hyper x,1$SQMATRIX(dim,ℚ)::ℒ ℒ ℚ)

\label{eq11}\left[ 1, \: i , \: j , \:{ij}\right](11)
Type: List(CaleyDickson?(CaleyDickson?(Expression(Integer),i,1),j,*01q(1)))
axiom
-- Multiplication table:
M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j, i,1..dim), j,1..dim)

\label{eq12}\left[ 
\begin{array}{cccc}
1 & i & j &{ij}
\
i & - 1 & -{ij}& j 
\
j &{ij}& -{q_{1}}&{-{q_{1}}i}
\
{ij}& - j &{{q_{1}}i}& -{q_{1}}
(12)
Type: Matrix(CaleyDickson?(CaleyDickson?(Expression(Integer),i,1),j,*01q(1)))
axiom
-- Function to divide the matrix entries by a basis element
S(y) == map(x +-> real real(x/y),M)
Type: Void
axiom
-- The result is a nested list
ѕ :=map(S,B)::ℒ ℒ ℒ ℚ
axiom
Compiling function S with type CaleyDickson(CaleyDickson(Expression(
      Integer),i,1),j,*01q(1)) -> Matrix(Expression(Integer))

\label{eq13}\begin{array}{@{}l}
\displaystyle
\left[{\left[{\left[ 1, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \: - 1, \: 0, \: 0 \right]}, \:{\left[ 0, \: 0, \: -{q_{1}}, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: -{q_{1}}\right]}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[{\left[ 0, \: 1, \: 0, \: 0 \right]}, \:{\left[ 1, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: -{q_{1}}\right]}, \:{\left[ 0, \: 0, \:{q_{1}}, \: 0 \right]}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[{\left[ 0, \: 0, \: 1, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: 1 \right]}, \:{\left[ 1, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \: - 1, \: 0, \: 0 \right]}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[{\left[ 0, \: 0, \: 0, \: 1 \right]}, \:{\left[ 0, \: 0, \: - 1, \: 0 \right]}, \:{\left[ 0, \: 1, \: 0, \: 0 \right]}, \:{\left[ 1, \: 0, \: 0, \: 0 \right]}\right]}\right] (13)
Type: List(List(List(Expression(Integer))))
axiom
-- structure constants form a tensor operator
Y := Σ(Σ(Σ(ѕ(i)(k)(j)*𝐞.i*𝐝.j*𝐝.k, i,1..dim), j,1..dim), k,1..dim)

\label{eq14}\begin{array}{@{}l}
\displaystyle
{|_{\  1}^{\  1 \  1}}+{|_{\  i}^{\  1 \  i}}+{|_{\  j}^{\  1 \  j}}+{|_{\  k}^{\  1 \  k}}+{|_{\  i}^{\  i \  1}}-{|_{\  1}^{\  i \  i}}+{|_{\  k}^{\  i \  j}}- 
\
\
\displaystyle
{|_{\  j}^{\  i \  k}}+{|_{\  j}^{\  j \  1}}-{|_{\  k}^{\  j \  i}}-{{q_{1}}\ {|_{\  1}^{\  j \  j}}}+{{q_{1}}\ {|_{\  i}^{\  j \  k}}}+{|_{\  k}^{\  k \  1}}+ 
\
\
\displaystyle
{|_{\  j}^{\  k \  i}}-{{q_{1}}\ {|_{\  i}^{\  k \  j}}}-{{q_{1}}\ {|_{\  1}^{\  k \  k}}}
(14)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
arity Y

\label{eq15}{+^2}\over +(15)
Type: ClosedProp?(ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer)))
axiom
matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y, i,1..dim), j,1..dim)

\label{eq16}\left[ 
\begin{array}{cccc}
{|_{\  1}}&{|_{\  i}}&{|_{\  j}}&{|_{\  k}}
\
{|_{\  i}}& -{|_{\  1}}& -{|_{\  k}}&{|_{\  j}}
\
{|_{\  j}}&{|_{\  k}}& -{{q_{1}}\ {|_{\  1}}}& -{{q_{1}}\ {|_{\  i}}}
\
{|_{\  k}}& -{|_{\  j}}&{{q_{1}}\ {|_{\  i}}}& -{{q_{1}}\ {|_{\  1}}}
(16)
Type: Matrix(ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer)))

Multiplication of arbitrary quaternions a and b

axiom
a:=Σ(sb('a,[i])*𝐞.i, i,1..dim)

\label{eq17}{{a_{1}}\ {|_{\  1}}}+{{a_{2}}\ {|_{\  i}}}+{{a_{3}}\ {|_{\  j}}}+{{a_{4}}\ {|_{\  k}}}(17)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
b:=Σ(sb('b,[i])*𝐞.i, i,1..dim)

\label{eq18}{{b_{1}}\ {|_{\  1}}}+{{b_{2}}\ {|_{\  i}}}+{{b_{3}}\ {|_{\  j}}}+{{b_{4}}\ {|_{\  k}}}(18)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
(a*b)/Y

\label{eq19}\begin{array}{@{}l}
\displaystyle
{{\left({{\left(-{{a_{4}}\ {b_{4}}}-{{a_{3}}\ {b_{3}}}\right)}\ {q_{1}}}-{{a_{2}}\ {b_{2}}}+{{a_{1}}\ {b_{1}}}\right)}\ {|_{\  1}}}+ 
\
\
\displaystyle
{{\left({{\left({{a_{3}}\ {b_{4}}}-{{a_{4}}\ {b_{3}}}\right)}\ {q_{1}}}+{{a_{1}}\ {b_{2}}}+{{a_{2}}\ {b_{1}}}\right)}\ {|_{\  i}}}+ 
\
\
\displaystyle
{{\left(-{{a_{2}}\ {b_{4}}}+{{a_{1}}\ {b_{3}}}+{{a_{4}}\ {b_{2}}}+{{a_{3}}\ {b_{1}}}\right)}\ {|_{\  j}}}+ 
\
\
\displaystyle
{{\left({{a_{1}}\ {b_{4}}}+{{a_{2}}\ {b_{3}}}-{{a_{3}}\ {b_{2}}}+{{a_{4}}\ {b_{1}}}\right)}\ {|_{\  k}}}
(19)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

Multiplication is Associative

axiom
test(
  ( I Y ) / _
  (  Y  ) = _
  ( Y I ) / _
  (  Y  ) )

\label{eq20} \mbox{\rm true} (20)
Type: Boolean

A scalar product is denoted by the (2,0)-tensor U = \{ u_{ij} \}

axiom
U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)

\label{eq21}\begin{array}{@{}l}
\displaystyle
{{u^{1, \: 1}}\ {|^{\  1 \  1}}}+{{u^{1, \: 2}}\ {|^{\  1 \  i}}}+{{u^{1, \: 3}}\ {|^{\  1 \  j}}}+{{u^{1, \: 4}}\ {|^{\  1 \  k}}}+ 
\
\
\displaystyle
{{u^{2, \: 1}}\ {|^{\  i \  1}}}+{{u^{2, \: 2}}\ {|^{\  i \  i}}}+{{u^{2, \: 3}}\ {|^{\  i \  j}}}+{{u^{2, \: 4}}\ {|^{\  i \  k}}}+{{u^{3, \: 1}}\ {|^{\  j \  1}}}+ 
\
\
\displaystyle
{{u^{3, \: 2}}\ {|^{\  j \  i}}}+{{u^{3, \: 3}}\ {|^{\  j \  j}}}+{{u^{3, \: 4}}\ {|^{\  j \  k}}}+{{u^{4, \: 1}}\ {|^{\  k \  1}}}+ 
\
\
\displaystyle
{{u^{4, \: 2}}\ {|^{\  k \  i}}}+{{u^{4, \: 3}}\ {|^{\  k \  j}}}+{{u^{4, \: 4}}\ {|^{\  k \  k}}}
(21)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

Definition 1

We say that the scalar product is associative if the tensor equation holds:

    Y   =   Y
     U     U

In other words, if the (3,0)-tensor:


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\label{eq22}
  \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}
  (22)
(three-point function) is zero.

Using the LinearOperator? domain in Axiom and some carefully chosen symbols we can easily enter expressions that are both readable and interpreted by Axiom as "graphical calculus" diagrams describing complex products and compositions of linear operators.

axiom
ω:𝐋 :=
     (    Y I    )  /
           U        -
     (    I Y    )  /
           U

\label{eq23}\begin{array}{@{}l}
\displaystyle
{{\left({u^{2, \: 1}}-{u^{1, \: 2}}\right)}\ {|^{\  1 \  i \  1}}}+{{\left({u^{2, \: 2}}+{u^{1, \: 1}}\right)}\ {|^{\  1 \  i \  i}}}+ 
\
\
\displaystyle
{{\left({u^{2, \: 3}}-{u^{1, \: 4}}\right)}\ {|^{\  1 \  i \  j}}}+{{\left({u^{2, \: 4}}+{u^{1, \: 3}}\right)}\ {|^{\  1 \  i \  k}}}+ 
\
\
\displaystyle
{{\left({u^{3, \: 1}}-{u^{1, \: 3}}\right)}\ {|^{\  1 \  j \  1}}}+{{\left({u^{3, \: 2}}+{u^{1, \: 4}}\right)}\ {|^{\  1 \  j \  i}}}+ 
\
\
\displaystyle
{{\left({{u^{1, \: 1}}\ {q_{1}}}+{u^{3, \: 3}}\right)}\ {|^{\  1 \  j \  j}}}+{{\left(-{{u^{1, \: 2}}\ {q_{1}}}+{u^{3, \: 4}}\right)}\ {|^{\  1 \  j \  k}}}+ 
\
\
\displaystyle
{{\left({u^{4, \: 1}}-{u^{1, \: 4}}\right)}\ {|^{\  1 \  k \  1}}}+{{\left({u^{4, \: 2}}-{u^{1, \: 3}}\right)}\ {|^{\  1 \  k \  i}}}+ 
\
\
\displaystyle
{{\left({{u^{1, \: 2}}\ {q_{1}}}+{u^{4, \: 3}}\right)}\ {|^{\  1 \  k \  j}}}+{{\left({{u^{1, \: 1}}\ {q_{1}}}+{u^{4, \: 4}}\right)}\ {|^{\  1 \  k \  k}}}+ 
\
\
\displaystyle
{{\left(-{u^{2, \: 2}}-{u^{1, \: 1}}\right)}\ {|^{\  i \  i \  1}}}+{{\left({u^{2, \: 1}}-{u^{1, \: 2}}\right)}\ {|^{\  i \  i \  i}}}+ 
\
\
\displaystyle
{{\left(-{u^{2, \: 4}}-{u^{1, \: 3}}\right)}\ {|^{\  i \  i \  j}}}+{{\left({u^{2, \: 3}}-{u^{1, \: 4}}\right)}\ {|^{\  i \  i \  k}}}+ 
\
\
\displaystyle
{{\left({u^{4, \: 1}}-{u^{2, \: 3}}\right)}\ {|^{\  i \  j \  1}}}+{{\left({u^{4, \: 2}}+{u^{2, \: 4}}\right)}\ {|^{\  i \  j \  i}}}+ 
\
\
\displaystyle
{{\left({{u^{2, \: 1}}\ {q_{1}}}+{u^{4, \: 3}}\right)}\ {|^{\  i \  j \  j}}}+{{\left(-{{u^{2, \: 2}}\ {q_{1}}}+{u^{4, \: 4}}\right)}\ {|^{\  i \  j \  k}}}+ 
\
\
\displaystyle
{{\left(-{u^{3, \: 1}}-{u^{2, \: 4}}\right)}\ {|^{\  i \  k \  1}}}+{{\left(-{u^{3, \: 2}}-{u^{2, \: 3}}\right)}\ {|^{\  i \  k \  i}}}+ 
\
\
\displaystyle
{{\left({{u^{2, \: 2}}\ {q_{1}}}-{u^{3, \: 3}}\right)}\ {|^{\  i \  k \  j}}}+{{\left({{u^{2, \: 1}}\ {q_{1}}}-{u^{3, \: 4}}\right)}\ {|^{\  i \  k \  k}}}+ 
\
\
\displaystyle
{{\left(-{u^{4, \: 1}}-{u^{3, \: 2}}\right)}\ {|^{\  j \  i \  1}}}+{{\left(-{u^{4, \: 2}}+{u^{3, \: 1}}\right)}\ {|^{\  j \  i \  i}}}+ 
\
\
\displaystyle
{{\left(-{u^{4, \: 3}}-{u^{3, \: 4}}\right)}\ {|^{\  j \  i \  j}}}+{{\left(-{u^{4, \: 4}}+{u^{3, \: 3}}\right)}\ {|^{\  j \  i \  k}}}+ 
\
\
\displaystyle
{{\left(-{{u^{1, \: 1}}\ {q_{1}}}-{u^{3, \: 3}}\right)}\ {|^{\  j \  j \  1}}}+{{\left(-{{u^{1, \: 2}}\ {q_{1}}}+{u^{3, \: 4}}\right)}\ {|^{\  j \  j \  i}}}+ 
\
\
\displaystyle
{{\left({u^{3, \: 1}}-{u^{1, \: 3}}\right)}\ {q_{1}}\ {|^{\  j \  j \  j}}}+{{\left(-{u^{3, \: 2}}-{u^{1, \: 4}}\right)}\ {q_{1}}\ {|^{\  j \  j \  k}}}+ 
\
\
\displaystyle
{{\left({{u^{2, \: 1}}\ {q_{1}}}-{u^{3, \: 4}}\right)}\ {|^{\  j \  k \  1}}}+{{\left({{u^{2, \: 2}}\ {q_{1}}}-{u^{3, \: 3}}\right)}\ {|^{\  j \  k \  i}}}+ 
\
\
\displaystyle
{{\left({u^{3, \: 2}}+{u^{2, \: 3}}\right)}\ {q_{1}}\ {|^{\  j \  k \  j}}}+{{\left({u^{3, \: 1}}+{u^{2, \: 4}}\right)}\ {q_{1}}\ {|^{\  j \  k \  k}}}+ 
\
\
\displaystyle
{{\left(-{u^{4, \: 2}}+{u^{3, \: 1}}\right)}\ {|^{\  k \  i \  1}}}+{{\left({u^{4, \: 1}}+{u^{3, \: 2}}\right)}\ {|^{\  k \  i \  i}}}+ 
\
\
\displaystyle
{{\left(-{u^{4, \: 4}}+{u^{3, \: 3}}\right)}\ {|^{\  k \  i \  j}}}+{{\left({u^{4, \: 3}}+{u^{3, \: 4}}\right)}\ {|^{\  k \  i \  k}}}+ 
\
\
\displaystyle
{{\left(-{{u^{2, \: 1}}\ {q_{1}}}-{u^{4, \: 3}}\right)}\ {|^{\  k \  j \  1}}}+{{\left(-{{u^{2, \: 2}}\ {q_{1}}}+{u^{4, \: 4}}\right)}\ {|^{\  k \  j \  i}}}+ 
\
\
\displaystyle
{{\left({u^{4, \: 1}}-{u^{2, \: 3}}\right)}\ {q_{1}}\ {|^{\  k \  j \  j}}}+{{\left(-{u^{4, \: 2}}-{u^{2, \: 4}}\right)}\ {q_{1}}\ {|^{\  k \  j \  k}}}+ 
\
\
\displaystyle
{{\left(-{{u^{1, \: 1}}\ {q_{1}}}-{u^{4, \: 4}}\right)}\ {|^{\  k \  k \  1}}}+{{\left(-{{u^{1, \: 2}}\ {q_{1}}}-{u^{4, \: 3}}\right)}\ {|^{\  k \  k \  i}}}+ 
\
\
\displaystyle
{{\left({u^{4, \: 2}}-{u^{1, \: 3}}\right)}\ {q_{1}}\ {|^{\  k \  k \  j}}}+{{\left({u^{4, \: 1}}-{u^{1, \: 4}}\right)}\ {q_{1}}\ {|^{\  k \  k \  k}}}
(23)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

Definition 2

An algebra with a non-degenerate associative scalar product is called a [Frobenius Algebra]?.

The Cartan-Killing Trace

axiom
Ú:=
   (  Y Λ  ) / _
   (   Y I ) / _
        V

\label{eq24}{4 \ {|^{\  1 \  1}}}-{4 \ {|^{\  i \  i}}}-{4 \ {q_{1}}\ {|^{\  j \  j}}}-{4 \ {q_{1}}\ {|^{\  k \  k}}}(24)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
Ù:=
   (  Λ Y  ) / _
   ( I Y   ) / _
      V

\label{eq25}{4 \ {|^{\  1 \  1}}}-{4 \ {|^{\  i \  i}}}-{4 \ {q_{1}}\ {|^{\  j \  j}}}-{4 \ {q_{1}}\ {|^{\  k \  k}}}(25)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
test(Ù=Ú)

\label{eq26} \mbox{\rm true} (26)
Type: Boolean

forms a non-degenerate associative scalar product for Y

axiom
Ũ := r*Ù

\label{eq27}{4 \  r \ {|^{\  1 \  1}}}-{4 \  r \ {|^{\  i \  i}}}-{4 \ {q_{1}}\  r \ {|^{\  j \  j}}}-{4 \ {q_{1}}\  r \ {|^{\  k \  k}}}(27)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
test
     (    Y I    )  /
           Ũ        =
     (    I Y    )  /
           Ũ

\label{eq28} \mbox{\rm true} (28)
Type: Boolean
axiom
determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ), j,1..dim), i,1..dim)

\label{eq29}-{{256}\ {{q_{1}}^2}\ {r^4}}(29)
Type: Expression(Integer)

General Solution

We may consider the problem where multiplication Y is given, and look for all associative scalar products U = U(Y)

This problem can be solved using linear algebra.

axiom
)expose MCALCFN
MultiVariableCalculusFunctions is now explicitly exposed in frame initial J := jacobian(ravel ω,concat map(variables,ravel U)::ℒ Symbol);
Type: Matrix(Expression(Integer))
axiom
u := transpose matrix [concat map(variables,ravel U)::ℒ Symbol];
Type: Matrix(Polynomial(Integer))
axiom
J::OutputForm * u::OutputForm = 0

\label{eq30}\begin{array}{@{}l}
\displaystyle
{{\left[ 
\begin{array}{cccccccccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & - 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & - 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \
{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 
\
0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 
\
0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 
\
0 &{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 
\
{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
- 1 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & - 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & - 1 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & - 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \
0 & 0 & 0 & 0 &{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 
\
0 & 0 & 0 & 0 & 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 &{q_{1}}& 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 &{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & - 1 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & - 1 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & - 1 
\
-{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 
\
0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 
\
0 & 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 &{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 &{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 &{q_{1}}& 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 &{q_{1}}& 0 & 0 &{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 &{q_{1}}&{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & - 1 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \
0 & 0 & 0 & 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 
\
0 & 0 & 0 & 0 & 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 
\
0 & 0 & 0 & 0 & 0 & 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 &{q_{1}}& 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & -{q_{1}}& 0 & 0 
\
-{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 
\
0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 
\
0 & 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{q_{1}}& 0 & 0 
\
0 & 0 & 0 & -{q_{1}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{q_{1}}& 0 & 0 & 0 
(30)
Type: Equation(OutputForm?)
axiom
nrows(J),ncols(J)

\label{eq31}\left[{64}, \:{16}\right](31)
Type: Tuple(PositiveInteger?)

The matrix J transforms the coefficients of the tensor U into coefficients of the tensor \Phi. We are looking for the general linear family of tensors U=U(Y,p_i) such that J transforms U into \Phi=0 for any such U.

If the null space of the J matrix is not empty we can use the basis to find all non-trivial solutions for U:

axiom
Ñ:=nullSpace(J)

\label{eq32}\begin{array}{@{}l}
\displaystyle
\left[{\left[ 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: 1, \: 0, \: 0, \: - 1, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0 \right]}, \: \right.
\
\
\displaystyle
\left.{\left[ 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: - 1, \: 1, \: 0, \: 0, \: 0, \: 0, \: 1, \: 0, \: 0 \right]}, \: \right.
\
\
\displaystyle
\left.{\left[ 0, \: -{1 \over{q_{1}}}, \: 0, \: 0, \: -{1 \over{q_{1}}}, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: - 1, \: 0, \: 0, \: 1, \: 0 \right]}, \: \right.
\
\
\displaystyle
\left.{\left[ -{1 \over{q_{1}}}, \: 0, \: 0, \: 0, \: 0, \:{1 \over{q_{1}}}, \: 0, \: 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 1 \right]}\right] 
(32)
Type: List(Vector(Expression(Integer)))
axiom
ℰ:=map((x,y)+->x=y, concat
       map(variables,ravel U), entries Σ(sb('p,[i])*Ñ.i, i,1..#Ñ) )

\label{eq33}\begin{array}{@{}l}
\displaystyle
\left[{{u^{1, \: 1}}= -{{p_{4}}\over{q_{1}}}}, \:{{u^{1, \: 2}}= -{{p_{3}}\over{q_{1}}}}, \:{{u^{1, \: 3}}={p_{2}}}, \:{{u^{1, \: 4}}={p_{1}}}, \: \right.
\
\
\displaystyle
\left.{{u^{2, \: 1}}= -{{p_{3}}\over{q_{1}}}}, \:{{u^{2, \: 2}}={{p_{4}}\over{q_{1}}}}, \:{{u^{2, \: 3}}={p_{1}}}, \:{{u^{2, \: 4}}= -{p_{2}}}, \: \right.
\
\
\displaystyle
\left.{{u^{3, \: 1}}={p_{2}}}, \:{{u^{3, \: 2}}= -{p_{1}}}, \:{{u^{3, \: 3}}={p_{4}}}, \:{{u^{3, \: 4}}= -{p_{3}}}, \: \right.
\
\
\displaystyle
\left.{{u^{4, \: 1}}={p_{1}}}, \:{{u^{4, \: 2}}={p_{2}}}, \:{{u^{4, \: 3}}={p_{3}}}, \:{{u^{4, \: 4}}={p_{4}}}\right] 
(33)
Type: List(Equation(Expression(Integer)))

This defines a family of pre-Frobenius algebras:

axiom
zero? eval(ω,ℰ)

\label{eq34} \mbox{\rm true} (34)
Type: Boolean

In general the pairing is not symmetric!

axiom
Ų:𝐋 := eval(U,ℰ)

\label{eq35}\begin{array}{@{}l}
\displaystyle
-{{{p_{4}}\over{q_{1}}}\ {|^{\  1 \  1}}}-{{{p_{3}}\over{q_{1}}}\ {|^{\  1 \  i}}}+{{p_{2}}\ {|^{\  1 \  j}}}+{{p_{1}}\ {|^{\  1 \  k}}}-{{{p_{3}}\over{q_{1}}}\ {|^{\  i \  1}}}+ 
\
\
\displaystyle
{{{p_{4}}\over{q_{1}}}\ {|^{\  i \  i}}}+{{p_{1}}\ {|^{\  i \  j}}}-{{p_{2}}\ {|^{\  i \  k}}}+{{p_{2}}\ {|^{\  j \  1}}}-{{p_{1}}\ {|^{\  j \  i}}}+{{p_{4}}\ {|^{\  j \  j}}}- 
\
\
\displaystyle
{{p_{3}}\ {|^{\  j \  k}}}+{{p_{1}}\ {|^{\  k \  1}}}+{{p_{2}}\ {|^{\  k \  i}}}+{{p_{3}}\ {|^{\  k \  j}}}+{{p_{4}}\ {|^{\  k \  k}}}
(35)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
matrix Ξ(Ξ((𝐞.i 𝐞.j)/Ų, i,1..dim), j,1..dim)

\label{eq36}\left[ 
\begin{array}{cccc}
-{{p_{4}}\over{q_{1}}}& -{{p_{3}}\over{q_{1}}}&{p_{2}}&{p_{1}}
\
-{{p_{3}}\over{q_{1}}}&{{p_{4}}\over{q_{1}}}& -{p_{1}}&{p_{2}}
\
{p_{2}}&{p_{1}}&{p_{4}}&{p_{3}}
\
{p_{1}}& -{p_{2}}& -{p_{3}}&{p_{4}}
(36)
Type: Matrix(ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer)))

This is the most general form of the "dot product" of two quaternions

axiom
(a*b)/Ų

\label{eq37}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{\left({{a_{4}}\ {b_{4}}}+{{a_{3}}\ {b_{3}}}\right)}\ {p_{4}}}+{{\left(-{{a_{3}}\ {b_{4}}}+{{a_{4}}\ {b_{3}}}\right)}\ {p_{3}}}+ 
\
\
\displaystyle
{{\left(-{{a_{2}}\ {b_{4}}}+{{a_{1}}\ {b_{3}}}+{{a_{4}}\ {b_{2}}}+{{a_{3}}\ {b_{1}}}\right)}\ {p_{2}}}+ 
\
\
\displaystyle
{{\left({{a_{1}}\ {b_{4}}}+{{a_{2}}\ {b_{3}}}-{{a_{3}}\ {b_{2}}}+{{a_{4}}\ {b_{1}}}\right)}\ {p_{1}}}
(37)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
(a*a)/Ų

\label{eq38}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({{\left({{a_{4}}^2}+{{a_{3}}^2}\right)}\ {p_{4}}}+{2 \ {a_{1}}\ {a_{3}}\ {p_{2}}}+{2 \ {a_{1}}\ {a_{4}}\ {p_{1}}}\right)}\ {q_{1}}}+ 
\
\
\displaystyle
{{\left({{a_{2}}^2}-{{a_{1}}^2}\right)}\ {p_{4}}}-{2 \ {a_{1}}\ {a_{2}}\ {p_{3}}}
(38)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

The scalar product must be non-degenerate:

axiom
Ů:=determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ų), j,1..dim), i,1..dim)

\label{eq39}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left(-{{p_{2}}^4}-{2 \ {{p_{1}}^2}\ {{p_{2}}^2}}-{{p_{1}}^4}\right)}\ {{q_{1}}^2}}+ 
\
\
\displaystyle
{{\left({{\left(-{2 \ {{p_{2}}^2}}-{2 \ {{p_{1}}^2}}\right)}\ {{p_{4}}^2}}+{{\left(-{2 \ {{p_{2}}^2}}-{2 \ {{p_{1}}^2}}\right)}\ {{p_{3}}^2}}\right)}\ {q_{1}}}- 
\
\
\displaystyle
{{p_{4}}^4}-{2 \ {{p_{3}}^2}\ {{p_{4}}^2}}-{{p_{3}}^4}
(39)
Type: Expression(Integer)
axiom
factor Ů

\label{eq40}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left(-{{p_{2}}^4}-{2 \ {{p_{1}}^2}\ {{p_{2}}^2}}-{{p_{1}}^4}\right)}\ {{q_{1}}^2}}+ 
\
\
\displaystyle
{{\left({{\left(-{2 \ {{p_{2}}^2}}-{2 \ {{p_{1}}^2}}\right)}\ {{p_{4}}^2}}+{{\left(-{2 \ {{p_{2}}^2}}-{2 \ {{p_{1}}^2}}\right)}\ {{p_{3}}^2}}\right)}\ {q_{1}}}- 
\
\
\displaystyle
{{p_{4}}^4}-{2 \ {{p_{3}}^2}\ {{p_{4}}^2}}-{{p_{3}}^4}
(40)
Type: Factored(Expression(Integer))

Definition 3

Co-pairing

Solve the [Snake Relation]? as a system of linear equations.

axiom
Ω:𝐋:=Σ(Σ(script('u,[[i,j]])*𝐞.i*𝐞.j, i,1..dim), j,1..dim)

\label{eq41}\begin{array}{@{}l}
\displaystyle
{{u_{1, \: 1}}\ {|_{\  1 \  1}}}+{{u_{1, \: 2}}\ {|_{\  1 \  i}}}+{{u_{1, \: 3}}\ {|_{\  1 \  j}}}+{{u_{1, \: 4}}\ {|_{\  1 \  k}}}+ 
\
\
\displaystyle
{{u_{2, \: 1}}\ {|_{\  i \  1}}}+{{u_{2, \: 2}}\ {|_{\  i \  i}}}+{{u_{2, \: 3}}\ {|_{\  i \  j}}}+{{u_{2, \: 4}}\ {|_{\  i \  k}}}+{{u_{3, \: 1}}\ {|_{\  j \  1}}}+ 
\
\
\displaystyle
{{u_{3, \: 2}}\ {|_{\  j \  i}}}+{{u_{3, \: 3}}\ {|_{\  j \  j}}}+{{u_{3, \: 4}}\ {|_{\  j \  k}}}+{{u_{4, \: 1}}\ {|_{\  k \  1}}}+ 
\
\
\displaystyle
{{u_{4, \: 2}}\ {|_{\  k \  i}}}+{{u_{4, \: 3}}\ {|_{\  k \  j}}}+{{u_{4, \: 4}}\ {|_{\  k \  k}}}
(41)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
d1:=(I*Ω)/(Ų*I);
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
d2:=(Ω*I)/(I*Ų);
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
equate(eq)==map((x,y)+->(x=y),ravel lhs eq, ravel rhs eq);
Type: Void
axiom
eq1:=equate(d1=I);
axiom
Compiling function equate with type Equation(ClosedLinearOperator(
      OrderedVariableList([1,i,j,k]),Expression(Integer))) -> List(
      Equation(Expression(Integer)))
Type: List(Equation(Expression(Integer)))
axiom
eq2:=equate(d2=I);
Type: List(Equation(Expression(Integer)))
axiom
snake:=solve(concat(eq1,eq2),concat Ξ(Ξ(script('u,[[i,j]]), i,1..dim), j,1..dim));
Type: List(List(Equation(Expression(Integer))))
axiom
if #snake ~= 1 then error "no solution"
Type: Void
axiom
Ω:=eval(Ω,snake(1))

\label{eq42}\begin{array}{@{}l}
\displaystyle
-{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  1}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  i}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  j}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  k}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  1}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  i}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  j}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  k}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  1}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  i}}}+ 
\
\
\displaystyle
{{{p_{4}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  j}}}+ 
\
\
\displaystyle
{{{p_{3}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  k}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  1}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  i}}}- 
\
\
\displaystyle
{{{p_{3}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  j}}}+ 
\
\
\displaystyle
{{{p_{4}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  k}}}
(42)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
matrix Ξ(Ξ(Ω/(𝐝.i*𝐝.j), i,1..dim), j,1..dim)

\label{eq43}\left[ 
\begin{array}{cccc}
-{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}& -{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}&{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}&{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}
\
-{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}&{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}&{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}& -{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}
\
{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}& -{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}&{{p_{4}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}& -{{p_{3}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}
\
{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}&{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}&{{p_{3}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}&{{p_{4}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}
(43)
Type: Matrix(ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer)))

Check "dimension" and the snake relations.

axiom
d:𝐋:=
       Ω    /
       X    /
       Ų

\label{eq44}4(44)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
test
    (    I Ω     )  /
    (     Ų I    )  =  I

\label{eq45} \mbox{\rm true} (45)
Type: Boolean
axiom
test
    (     Ω I    )  /
    (    I Ų     )  =  I

\label{eq46} \mbox{\rm true} (46)
Type: Boolean

Definition 4

Co-algebra

Compute the "three-point" function and use it to define co-multiplication.

axiom
W:=(Y,I)/Ų

\label{eq47}\begin{array}{@{}l}
\displaystyle
-{{{p_{4}}\over{q_{1}}}\ {|^{\  1 \  1 \  1}}}-{{{p_{3}}\over{q_{1}}}\ {|^{\  1 \  1 \  i}}}+{{p_{2}}\ {|^{\  1 \  1 \  j}}}+{{p_{1}}\ {|^{\  1 \  1 \  k}}}- 
\
\
\displaystyle
{{{p_{3}}\over{q_{1}}}\ {|^{\  1 \  i \  1}}}+{{{p_{4}}\over{q_{1}}}\ {|^{\  1 \  i \  i}}}+{{p_{1}}\ {|^{\  1 \  i \  j}}}-{{p_{2}}\ {|^{\  1 \  i \  k}}}+{{p_{2}}\ {|^{\  1 \  j \  1}}}- 
\
\
\displaystyle
{{p_{1}}\ {|^{\  1 \  j \  i}}}+{{p_{4}}\ {|^{\  1 \  j \  j}}}-{{p_{3}}\ {|^{\  1 \  j \  k}}}+{{p_{1}}\ {|^{\  1 \  k \  1}}}+{{p_{2}}\ {|^{\  1 \  k \  i}}}+ 
\
\
\displaystyle
{{p_{3}}\ {|^{\  1 \  k \  j}}}+{{p_{4}}\ {|^{\  1 \  k \  k}}}-{{{p_{3}}\over{q_{1}}}\ {|^{\  i \  1 \  1}}}+{{{p_{4}}\over{q_{1}}}\ {|^{\  i \  1 \  i}}}+{{p_{1}}\ {|^{\  i \  1 \  j}}}- 
\
\
\displaystyle
{{p_{2}}\ {|^{\  i \  1 \  k}}}+{{{p_{4}}\over{q_{1}}}\ {|^{\  i \  i \  1}}}+{{{p_{3}}\over{q_{1}}}\ {|^{\  i \  i \  i}}}-{{p_{2}}\ {|^{\  i \  i \  j}}}-{{p_{1}}\ {|^{\  i \  i \  k}}}+ 
\
\
\displaystyle
{{p_{1}}\ {|^{\  i \  j \  1}}}+{{p_{2}}\ {|^{\  i \  j \  i}}}+{{p_{3}}\ {|^{\  i \  j \  j}}}+{{p_{4}}\ {|^{\  i \  j \  k}}}-{{p_{2}}\ {|^{\  i \  k \  1}}}+ 
\
\
\displaystyle
{{p_{1}}\ {|^{\  i \  k \  i}}}-{{p_{4}}\ {|^{\  i \  k \  j}}}+{{p_{3}}\ {|^{\  i \  k \  k}}}+{{p_{2}}\ {|^{\  j \  1 \  1}}}-{{p_{1}}\ {|^{\  j \  1 \  i}}}+ 
\
\
\displaystyle
{{p_{4}}\ {|^{\  j \  1 \  j}}}-{{p_{3}}\ {|^{\  j \  1 \  k}}}-{{p_{1}}\ {|^{\  j \  i \  1}}}-{{p_{2}}\ {|^{\  j \  i \  i}}}-{{p_{3}}\ {|^{\  j \  i \  j}}}- 
\
\
\displaystyle
{{p_{4}}\ {|^{\  j \  i \  k}}}+{{p_{4}}\ {|^{\  j \  j \  1}}}+{{p_{3}}\ {|^{\  j \  j \  i}}}-{{p_{2}}\ {q_{1}}\ {|^{\  j \  j \  j}}}- 
\
\
\displaystyle
{{p_{1}}\ {q_{1}}\ {|^{\  j \  j \  k}}}-{{p_{3}}\ {|^{\  j \  k \  1}}}+{{p_{4}}\ {|^{\  j \  k \  i}}}+{{p_{1}}\ {q_{1}}\ {|^{\  j \  k \  j}}}- 
\
\
\displaystyle
{{p_{2}}\ {q_{1}}\ {|^{\  j \  k \  k}}}+{{p_{1}}\ {|^{\  k \  1 \  1}}}+{{p_{2}}\ {|^{\  k \  1 \  i}}}+{{p_{3}}\ {|^{\  k \  1 \  j}}}+ 
\
\
\displaystyle
{{p_{4}}\ {|^{\  k \  1 \  k}}}+{{p_{2}}\ {|^{\  k \  i \  1}}}-{{p_{1}}\ {|^{\  k \  i \  i}}}+{{p_{4}}\ {|^{\  k \  i \  j}}}-{{p_{3}}\ {|^{\  k \  i \  k}}}+ 
\
\
\displaystyle
{{p_{3}}\ {|^{\  k \  j \  1}}}-{{p_{4}}\ {|^{\  k \  j \  i}}}-{{p_{1}}\ {q_{1}}\ {|^{\  k \  j \  j}}}+{{p_{2}}\ {q_{1}}\ {|^{\  k \  j \  k}}}+ 
\
\
\displaystyle
{{p_{4}}\ {|^{\  k \  k \  1}}}+{{p_{3}}\ {|^{\  k \  k \  i}}}-{{p_{2}}\ {q_{1}}\ {|^{\  k \  k \  j}}}-{{p_{1}}\ {q_{1}}\ {|^{\  k \  k \  k}}}
(47)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
λ:=(Ω,I,Ω)/(I,W,I)

\label{eq48}\begin{array}{@{}l}
\displaystyle
-{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  1}^{\  1}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  i}^{\  1}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  j}^{\  1}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  k}^{\  1}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  1}^{\  1}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  i}^{\  1}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  j}^{\  1}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  k}^{\  1}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  1}^{\  1}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  i}^{\  1}}}+ 
\
\
\displaystyle
{{{p_{4}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  j}^{\  1}}}+ 
\
\
\displaystyle
{{{p_{3}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  k}^{\  1}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  1}^{\  1}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  i}^{\  1}}}- 
\
\
\displaystyle
{{{p_{3}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  j}^{\  1}}}+ 
\
\
\displaystyle
{{{p_{4}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  k}^{\  1}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  1}^{\  i}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  i}^{\  i}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  j}^{\  i}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  k}^{\  i}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  1}^{\  i}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  i}^{\  i}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  j}^{\  i}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  k}^{\  i}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  1}^{\  i}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  i}^{\  i}}}+ 
\
\
\displaystyle
{{{p_{3}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  j}^{\  i}}}- 
\
\
\displaystyle
{{{p_{4}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  k}^{\  i}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  1}^{\  i}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  i}^{\  i}}}+ 
\
\
\displaystyle
{{{p_{4}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  j}^{\  i}}}+ 
\
\
\displaystyle
{{{p_{3}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  k}^{\  i}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  1}^{\  j}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  i}^{\  j}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  j}^{\  j}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  k}^{\  j}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  1}^{\  j}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  i}^{\  j}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  j}^{\  j}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  k}^{\  j}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  1}^{\  j}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  i}^{\  j}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  j}^{\  j}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  k}^{\  j}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  1}^{\  j}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  i}^{\  j}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  j}^{\  j}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  k}^{\  j}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  1}^{\  k}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  i}^{\  k}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  j}^{\  k}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  k}^{\  k}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  1}^{\  k}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  i}^{\  k}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  j}^{\  k}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  k}^{\  k}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  1}^{\  k}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  i}^{\  k}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  j}^{\  k}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  k}^{\  k}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  1}^{\  k}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  i}^{\  k}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  j}^{\  k}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  k}^{\  k}}}
(48)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

axiom
test
     (    I Ω     )  /
     (     Y I    )  =  λ

\label{eq49} \mbox{\rm true} (49)
Type: Boolean
axiom
test
     (     Ω I    )  /
     (    I Y     )  =  λ

\label{eq50} \mbox{\rm true} (50)
Type: Boolean

Co-associativity

axiom
test(
  (  λ  ) / _
  ( I λ ) = _
  (  λ  ) / _
  ( λ I ) )

\label{eq51} \mbox{\rm true} (51)
Type: Boolean

Frobenius Condition

axiom
H :=
         Y    /
         λ

\label{eq52}\begin{array}{@{}l}
\displaystyle
-{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  1}^{\  1 \  1}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  i}^{\  1 \  1}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  j}^{\  1 \  1}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  k}^{\  1 \  1}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  1}^{\  1 \  1}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  i}^{\  1 \  1}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  j}^{\  1 \  1}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  k}^{\  1 \  1}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  1}^{\  1 \  1}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  i}^{\  1 \  1}}}+ 
\
\
\displaystyle
{{{p_{4}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  j}^{\  1 \  1}}}+ 
\
\
\displaystyle
{{{p_{3}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  k}^{\  1 \  1}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  1}^{\  1 \  1}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  i}^{\  1 \  1}}}- 
\
\
\displaystyle
{{{p_{3}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  j}^{\  1 \  1}}}+ 
\
\
\displaystyle
{{{p_{4}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  k}^{\  1 \  1}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  1}^{\  1 \  i}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  i}^{\  1 \  i}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  j}^{\  1 \  i}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  k}^{\  1 \  i}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  1}^{\  1 \  i}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  i}^{\  1 \  i}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  j}^{\  1 \  i}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  k}^{\  1 \  i}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  1}^{\  1 \  i}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  i}^{\  1 \  i}}}+ 
\
\
\displaystyle
{{{p_{3}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  j}^{\  1 \  i}}}- 
\
\
\displaystyle
{{{p_{4}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  k}^{\  1 \  i}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  1}^{\  1 \  i}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  i}^{\  1 \  i}}}+ 
\
\
\displaystyle
{{{p_{4}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  j}^{\  1 \  i}}}+ 
\
\
\displaystyle
{{{p_{3}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  k}^{\  1 \  i}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  1}^{\  1 \  j}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  i}^{\  1 \  j}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  j}^{\  1 \  j}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  k}^{\  1 \  j}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  1}^{\  1 \  j}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  i}^{\  1 \  j}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  j}^{\  1 \  j}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  k}^{\  1 \  j}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  1}^{\  1 \  j}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  i}^{\  1 \  j}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  j}^{\  1 \  j}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  k}^{\  1 \  j}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  1}^{\  1 \  j}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  i}^{\  1 \  j}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  j}^{\  1 \  j}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  k}^{\  1 \  j}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  1}^{\  1 \  k}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  i}^{\  1 \  k}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  j}^{\  1 \  k}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  k}^{\  1 \  k}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  1}^{\  1 \  k}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  i}^{\  1 \  k}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  j}^{\  1 \  k}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  k}^{\  1 \  k}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  1}^{\  1 \  k}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  i}^{\  1 \  k}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  j}^{\  1 \  k}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  k}^{\  1 \  k}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  1}^{\  1 \  k}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  i}^{\  1 \  k}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  j}^{\  1 \  k}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  k}^{\  1 \  k}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  1}^{\  i \  1}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  i}^{\  i \  1}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  j}^{\  i \  1}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  k}^{\  i \  1}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  1}^{\  i \  1}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  i}^{\  i \  1}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  j}^{\  i \  1}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  k}^{\  i \  1}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  1}^{\  i \  1}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  i}^{\  i \  1}}}+ 
\
\
\displaystyle
{{{p_{3}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  j}^{\  i \  1}}}- 
\
\
\displaystyle
{{{p_{4}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  k}^{\  i \  1}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  1}^{\  i \  1}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  i}^{\  i \  1}}}+ 
\
\
\displaystyle
{{{p_{4}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  j}^{\  i \  1}}}+ 
\
\
\displaystyle
{{{p_{3}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  k}^{\  i \  1}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  1}^{\  i \  i}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  i}^{\  i \  i}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  j}^{\  i \  i}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  k}^{\  i \  i}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  1}^{\  i \  i}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  i}^{\  i \  i}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  j}^{\  i \  i}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  k}^{\  i \  i}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  1}^{\  i \  i}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  i}^{\  i \  i}}}- 
\
\
\displaystyle
{{{p_{4}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  j}^{\  i \  i}}}- 
\
\
\displaystyle
{{{p_{3}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  k}^{\  i \  i}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  1}^{\  i \  i}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  i}^{\  i \  i}}}+ 
\
\
\displaystyle
{{{p_{3}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  j}^{\  i \  i}}}- 
\
\
\displaystyle
{{{p_{4}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  k}^{\  i \  i}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  1}^{\  i \  j}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  i}^{\  i \  j}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  j}^{\  i \  j}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  k}^{\  i \  j}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  1}^{\  i \  j}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  i}^{\  i \  j}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  j}^{\  i \  j}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  k}^{\  i \  j}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  1}^{\  i \  j}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  i}^{\  i \  j}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  j}^{\  i \  j}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  k}^{\  i \  j}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  1}^{\  i \  j}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  i}^{\  i \  j}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  j}^{\  i \  j}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  k}^{\  i \  j}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  1}^{\  i \  k}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  i}^{\  i \  k}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  j}^{\  i \  k}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  k}^{\  i \  k}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  1}^{\  i \  k}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  i}^{\  i \  k}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  j}^{\  i \  k}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  k}^{\  i \  k}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  1}^{\  i \  k}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  i}^{\  i \  k}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  j}^{\  i \  k}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  k}^{\  i \  k}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  1}^{\  i \  k}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  i}^{\  i \  k}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  j}^{\  i \  k}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  k}^{\  i \  k}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  1}^{\  j \  1}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  i}^{\  j \  1}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  j}^{\  j \  1}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  k}^{\  j \  1}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  1}^{\  j \  1}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  i}^{\  j \  1}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  j}^{\  j \  1}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  k}^{\  j \  1}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  1}^{\  j \  1}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  i}^{\  j \  1}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  j}^{\  j \  1}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  k}^{\  j \  1}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  1}^{\  j \  1}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  i}^{\  j \  1}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  j}^{\  j \  1}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  k}^{\  j \  1}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  1}^{\  j \  i}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  i}^{\  j \  i}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  j}^{\  j \  i}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  k}^{\  j \  i}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  1}^{\  j \  i}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  i}^{\  j \  i}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  j}^{\  j \  i}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  k}^{\  j \  i}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  1}^{\  j \  i}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  i}^{\  j \  i}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  j}^{\  j \  i}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  k}^{\  j \  i}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  1}^{\  j \  i}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  i}^{\  j \  i}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  j}^{\  j \  i}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  k}^{\  j \  i}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  1}^{\  j \  j}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  i}^{\  j \  j}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  j}^{\  j \  j}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  k}^{\  j \  j}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  1}^{\  j \  j}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  i}^{\  j \  j}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  j}^{\  j \  j}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  k}^{\  j \  j}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  1}^{\  j \  j}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  i}^{\  j \  j}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  j}^{\  j \  j}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  k}^{\  j \  j}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  1}^{\  j \  j}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  i}^{\  j \  j}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  j}^{\  j \  j}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  k}^{\  j \  j}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  1}^{\  j \  k}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  i}^{\  j \  k}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  j}^{\  j \  k}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  k}^{\  j \  k}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  1}^{\  j \  k}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  i}^{\  j \  k}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  j}^{\  j \  k}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  k}^{\  j \  k}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  1}^{\  j \  k}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  i}^{\  j \  k}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  j}^{\  j \  k}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  k}^{\  j \  k}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  1}^{\  j \  k}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  i}^{\  j \  k}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  j}^{\  j \  k}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  k}^{\  j \  k}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  1}^{\  k \  1}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  i}^{\  k \  1}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  j}^{\  k \  1}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  k}^{\  k \  1}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  1}^{\  k \  1}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  i}^{\  k \  1}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  j}^{\  k \  1}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  k}^{\  k \  1}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  1}^{\  k \  1}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  i}^{\  k \  1}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  j}^{\  k \  1}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  k}^{\  k \  1}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  1}^{\  k \  1}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  i}^{\  k \  1}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  j}^{\  k \  1}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  k}^{\  k \  1}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  1}^{\  k \  i}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  i}^{\  k \  i}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  j}^{\  k \  i}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  k}^{\  k \  i}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  1}^{\  k \  i}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  i}^{\  k \  i}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  j}^{\  k \  i}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  k}^{\  k \  i}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  1}^{\  k \  i}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  i}^{\  k \  i}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  j}^{\  k \  i}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  k}^{\  k \  i}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  1}^{\  k \  i}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  i}^{\  k \  i}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  j}^{\  k \  i}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  k}^{\  k \  i}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  1}^{\  k \  j}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  i}^{\  k \  j}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  j}^{\  k \  j}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  k}^{\  k \  j}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  1}^{\  k \  j}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  i}^{\  k \  j}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  j}^{\  k \  j}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  k}^{\  k \  j}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  1}^{\  k \  j}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  i}^{\  k \  j}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  j}^{\  k \  j}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  k}^{\  k \  j}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  1}^{\  k \  j}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  i}^{\  k \  j}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  j}^{\  k \  j}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  k}^{\  k \  j}}}+ 
\
\
\displaystyle
{{{{p_{4}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  1}^{\  k \  k}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  i}^{\  k \  k}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  j}^{\  k \  k}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1 \  k}^{\  k \  k}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  1}^{\  k \  k}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  i}^{\  k \  k}}}+ 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  j}^{\  k \  k}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i \  k}^{\  k \  k}}}- 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  1}^{\  k \  k}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  i}^{\  k \  k}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  j}^{\  k \  k}}}- 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j \  k}^{\  k \  k}}}- 
\
\
\displaystyle
{{{{p_{1}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  1}^{\  k \  k}}}+ 
\
\
\displaystyle
{{{{p_{2}}\ {{q_{1}}^2}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  i}^{\  k \  k}}}+ 
\
\
\displaystyle
{{{{p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  j}^{\  k \  k}}}- 
\
\
\displaystyle
{{{{p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k \  k}^{\  k \  k}}}
(52)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
test
     (   λ I   )  /
     (  I Y    )  =  H

\label{eq53} \mbox{\rm true} (53)
Type: Boolean
axiom
test
     (   I λ   )  /
     (    Y I  )  =  H

\label{eq54} \mbox{\rm true} (54)
Type: Boolean

Bi-algebra conditions

axiom
ΦΦ:=         _
  (  λ λ  ) / _
  ( I X I ) / _
  (  Y Y  ) ;
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
test( ΦΦ=H )

\label{eq55} \mbox{\rm false} (55)
Type: Boolean
axiom
test( ΦΦ=H/H )

\label{eq56} \mbox{\rm false} (56)
Type: Boolean
axiom
solve(equate(ΦΦ=H),Ξ(sb('p,[i]), i,1..#Ñ))

\label{eq57}\left[ \right](57)
Type: List(List(Equation(Expression(Integer))))

i = Unit of the algebra

axiom
i:=𝐞.1

\label{eq58}|_{\  1}(58)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
test
         i     /
         λ     =    Ω

\label{eq59} \mbox{\rm true} (59)
Type: Boolean

Handle

axiom
Φ:𝐋 :=
         λ     /
         X     /
         Y

\label{eq60}\begin{array}{@{}l}
\displaystyle
-{{{4 \ {p_{4}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  1}^{\  1}}}- 
\
\
\displaystyle
{{{4 \ {p_{3}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  i}^{\  1}}}+ 
\
\
\displaystyle
{{{4 \ {p_{2}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  j}^{\  1}}}+ 
\
\
\displaystyle
{{{4 \ {p_{1}}\ {q_{1}}}\over{{{\left({{p_{2}}^2}+{{p_{1}}^2}\right)}\ {q_{1}}}+{{p_{4}}^2}+{{p_{3}}^2}}}\ {|_{\  k}^{\  1}}}
(60)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

Definition 5

Co-unit
  i 
  U
  

axiom
ι:𝐋:=
    (    i I   ) /
          Ų

\label{eq61}-{{{p_{4}}\over{q_{1}}}\ {|^{\  1}}}-{{{p_{3}}\over{q_{1}}}\ {|^{\  i}}}+{{p_{2}}\ {|^{\  j}}}+{{p_{1}}\ {|^{\  k}}}(61)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

Y=U
ι  
axiom
test
        Y     /
        ι     =  Ų

\label{eq62} \mbox{\rm true} (62)
Type: Boolean

For example:

axiom
ex1:=[q[1]=1,p[1]=1,p[2]=1,p[3]=1,p[4]=1]

\label{eq63}\left[{{q_{1}}= 1}, \:{{p_{1}}= 1}, \:{{p_{2}}= 1}, \:{{p_{3}}= 1}, \:{{p_{4}}= 1}\right](63)
Type: List(Equation(Polynomial(Integer)))
axiom
Ų0:𝐋  :=eval(Ų,ex1)

\label{eq64}\begin{array}{@{}l}
\displaystyle
-{|^{\  1 \  1}}-{|^{\  1 \  i}}+{|^{\  1 \  j}}+{|^{\  1 \  k}}-{|^{\  i \  1}}+{|^{\  i \  i}}+{|^{\  i \  j}}-{|^{\  i \  k}}+ 
\
\
\displaystyle
{|^{\  j \  1}}-{|^{\  j \  i}}+{|^{\  j \  j}}-{|^{\  j \  k}}+{|^{\  k \  1}}+{|^{\  k \  i}}+{|^{\  k \  j}}+{|^{\  k \  k}}
(64)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
Ω0:𝐋  :=eval(Ω,ex1)$𝐋

\label{eq65}\begin{array}{@{}l}
\displaystyle
-{{1 \over 4}\ {|_{\  1 \  1}}}-{{1 \over 4}\ {|_{\  1 \  i}}}+{{1 \over 4}\ {|_{\  1 \  j}}}+{{1 \over 4}\ {|_{\  1 \  k}}}-{{1 \over 4}\ {|_{\  i \  1}}}+{{1 \over 4}\ {|_{\  i \  i}}}- 
\
\
\displaystyle
{{1 \over 4}\ {|_{\  i \  j}}}+{{1 \over 4}\ {|_{\  i \  k}}}+{{1 \over 4}\ {|_{\  j \  1}}}+{{1 \over 4}\ {|_{\  j \  i}}}+{{1 \over 4}\ {|_{\  j \  j}}}+{{1 \over 4}\ {|_{\  j \  k}}}+ 
\
\
\displaystyle
{{1 \over 4}\ {|_{\  k \  1}}}-{{1 \over 4}\ {|_{\  k \  i}}}-{{1 \over 4}\ {|_{\  k \  j}}}+{{1 \over 4}\ {|_{\  k \  k}}}
(65)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
λ0:𝐋  :=eval(λ,ex1)$𝐋

\label{eq66}\begin{array}{@{}l}
\displaystyle
-{{1 \over 4}\ {|_{\  1 \  1}^{\  1}}}-{{1 \over 4}\ {|_{\  1 \  i}^{\  1}}}+{{1 \over 4}\ {|_{\  1 \  j}^{\  1}}}+{{1 \over 4}\ {|_{\  1 \  k}^{\  1}}}-{{1 \over 4}\ {|_{\  i \  1}^{\  1}}}+ 
\
\
\displaystyle
{{1 \over 4}\ {|_{\  i \  i}^{\  1}}}-{{1 \over 4}\ {|_{\  i \  j}^{\  1}}}+{{1 \over 4}\ {|_{\  i \  k}^{\  1}}}+{{1 \over 4}\ {|_{\  j \  1}^{\  1}}}+{{1 \over 4}\ {|_{\  j \  i}^{\  1}}}+ 
\
\
\displaystyle
{{1 \over 4}\ {|_{\  j \  j}^{\  1}}}+{{1 \over 4}\ {|_{\  j \  k}^{\  1}}}+{{1 \over 4}\ {|_{\  k \  1}^{\  1}}}-{{1 \over 4}\ {|_{\  k \  i}^{\  1}}}-{{1 \over 4}\ {|_{\  k \  j}^{\  1}}}+ 
\
\
\displaystyle
{{1 \over 4}\ {|_{\  k \  k}^{\  1}}}+{{1 \over 4}\ {|_{\  1 \  1}^{\  i}}}-{{1 \over 4}\ {|_{\  1 \  i}^{\  i}}}+{{1 \over 4}\ {|_{\  1 \  j}^{\  i}}}-{{1 \over 4}\ {|_{\  1 \  k}^{\  i}}}- 
\
\
\displaystyle
{{1 \over 4}\ {|_{\  i \  1}^{\  i}}}-{{1 \over 4}\ {|_{\  i \  i}^{\  i}}}+{{1 \over 4}\ {|_{\  i \  j}^{\  i}}}+{{1 \over 4}\ {|_{\  i \  k}^{\  i}}}-{{1 \over 4}\ {|_{\  j \  1}^{\  i}}}+ 
\
\
\displaystyle
{{1 \over 4}\ {|_{\  j \  i}^{\  i}}}+{{1 \over 4}\ {|_{\  j \  j}^{\  i}}}-{{1 \over 4}\ {|_{\  j \  k}^{\  i}}}+{{1 \over 4}\ {|_{\  k \  1}^{\  i}}}+{{1 \over 4}\ {|_{\  k \  i}^{\  i}}}+ 
\
\
\displaystyle
{{1 \over 4}\ {|_{\  k \  j}^{\  i}}}+{{1 \over 4}\ {|_{\  k \  k}^{\  i}}}-{{1 \over 4}\ {|_{\  1 \  1}^{\  j}}}-{{1 \over 4}\ {|_{\  1 \  i}^{\  j}}}-{{1 \over 4}\ {|_{\  1 \  j}^{\  j}}}- 
\
\
\displaystyle
{{1 \over 4}\ {|_{\  1 \  k}^{\  j}}}+{{1 \over 4}\ {|_{\  i \  1}^{\  j}}}-{{1 \over 4}\ {|_{\  i \  i}^{\  j}}}-{{1 \over 4}\ {|_{\  i \  j}^{\  j}}}+{{1 \over 4}\ {|_{\  i \  k}^{\  j}}}- 
\
\
\displaystyle
{{1 \over 4}\ {|_{\  j \  1}^{\  j}}}-{{1 \over 4}\ {|_{\  j \  i}^{\  j}}}+{{1 \over 4}\ {|_{\  j \  j}^{\  j}}}+{{1 \over 4}\ {|_{\  j \  k}^{\  j}}}+{{1 \over 4}\ {|_{\  k \  1}^{\  j}}}- 
\
\
\displaystyle
{{1 \over 4}\ {|_{\  k \  i}^{\  j}}}+{{1 \over 4}\ {|_{\  k \  j}^{\  j}}}-{{1 \over 4}\ {|_{\  k \  k}^{\  j}}}-{{1 \over 4}\ {|_{\  1 \  1}^{\  k}}}+{{1 \over 4}\ {|_{\  1 \  i}^{\  k}}}+ 
\
\
\displaystyle
{{1 \over 4}\ {|_{\  1 \  j}^{\  k}}}-{{1 \over 4}\ {|_{\  1 \  k}^{\  k}}}-{{1 \over 4}\ {|_{\  i \  1}^{\  k}}}-{{1 \over 4}\ {|_{\  i \  i}^{\  k}}}-{{1 \over 4}\ {|_{\  i \  j}^{\  k}}}- 
\
\
\displaystyle
{{1 \over 4}\ {|_{\  i \  k}^{\  k}}}-{{1 \over 4}\ {|_{\  j \  1}^{\  k}}}+{{1 \over 4}\ {|_{\  j \  i}^{\  k}}}-{{1 \over 4}\ {|_{\  j \  j}^{\  k}}}+{{1 \over 4}\ {|_{\  j \  k}^{\  k}}}- 
\
\
\displaystyle
{{1 \over 4}\ {|_{\  k \  1}^{\  k}}}-{{1 \over 4}\ {|_{\  k \  i}^{\  k}}}+{{1 \over 4}\ {|_{\  k \  j}^{\  k}}}+{{1 \over 4}\ {|_{\  k \  k}^{\  k}}}
(66)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
axiom
Φ0:𝐋 :=eval(Φ,ex1)$𝐋

\label{eq67}-{|_{\  1}^{\  1}}-{|_{\  i}^{\  1}}+{|_{\  j}^{\  1}}+{|_{\  k}^{\  1}}(67)
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))