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	SandBox Quaternion Algebra is Frobenius in Many Ways
	↑	→ Snake Relation

SandBoxPauliAlgebra

Edit detail for SandBoxPauliAlgebra revision 5 of 7

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Editor: Bill Page Time: 2011/06/03 18:56:05 GMT-7
Note:

changed:
-Now generate structure constants for Quaternion Algebra
Now generate structure constants for Pauli Algebra

added:
Units
\begin{axiom}
e:=𝐞.1; i:=𝐞.2; j:=𝐞.3; k:=𝐞.4; ij:=𝐞.5; ik:=𝐞.6; jk:=𝐞.7; ijk:=𝐞.8;
\end{axiom}

Multiplication of arbitrary quaternions $a$ and $b$
\begin{axiom}
a:=Σ(sb('a,[i])*𝐞.i, i,1..dim)
b:=Σ(sb('b,[i])*𝐞.i, i,1..dim)
(a*b)/Y
\end{axiom}


added:
$U = \{ u_{ij} \}$

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

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


changed:
-\end{axiom}
determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ), j,1..dim), i,1..dim)
\end{axiom}

General Solution

Frobenius Form (co-unit)
\begin{axiom}
d:=ε1*𝐝.1+εi*𝐝.2+εj*𝐝.3+εk*𝐝.4+εij*𝐝.5+εik*𝐝.6+εjk*𝐝.7+εijk*𝐝.8
Ų:= Y/d
\end{axiom}

In general the pairing is not symmetric!
\begin{axiom}
u1:=matrix Ξ(Ξ(retract((𝐞.i 𝐞.j)/Ų), i,1..dim), j,1..dim)
--eigenvectors(u1::Matrix FRAC POLY INT)
\end{axiom}

changed:
-Ů:=determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ), j,1..dim), i,1..dim)
-factor(numer Ů)/factor(denom Ů)
-\end{axiom}
--Ů:=determinant u1
--factor(numer Ů)/factor(denom Ů)
1
\end{axiom}

Cartan-Killing is a special case
\begin{axiom}
ck:=solve(equate(Ũ=Ų),[ε1,εi,εj,εk,εij,εik,εjk,εijk]).1
\end{axiom}

Frobenius scalar product of "vector" quaternions $a$ and $b$
\begin{axiom}
a:=sb('a,[1])*i+sb('a,[2])*j+sb('a,[3])*k
b:=sb('b,[1])*i+sb('b,[2])*j+sb('b,[3])*k
(a,a)/Ų
(b,b)/Ų
(a,b)/Ų
\end{axiom}

changed:
-  Co-scalar product (pairing)
  Co-scalar product

changed:
-mU:=inverse matrix Ξ(Ξ(retract((𝐞.i*𝐞.j)/Ũ), i,1..dim), j,1..dim)
mU:=inverse matrix Ξ(Ξ(retract((𝐞.i*𝐞.j)/Ų), i,1..dim), j,1..dim);

added:
The common demoninator is $1/\sqrt{\mathring{U}}$
\begin{axiom}
--squareFreePart factor denom Ů / squareFreePart factor numer Ů
matrix Ξ(Ξ(numer retract(Ω/(𝐝.i*𝐝.j)), i,1..dim), j,1..dim)
\end{axiom}


changed:
-d:𝐋:=
-       Ω    /
-       Ũ
O:𝐋:= Ω / Ų

changed:
-    (     Ũ I     )  =  I
    (     Ų I     )  =  I

changed:
-    (    I Ũ      )  =  I
-
-\end{axiom}
    (    I Ų      )  =  I

\end{axiom}

Cartan-Killing co-scalar
\begin{axiom}
eval(Ω,ck)
\end{axiom}

changed:
-W:=(Y I) / Ũ;
-
-\end{axiom}
-
-Cartan-Killing co-multiplication
-\begin{axiom}
-
-λ:=                     _
-     (    I ΩX     ) /  _
-     (     Y I     ) ;
W:= (Y I) / Ų;

  (  ΩX I ΩX  ) /
  (  I  W  I  )

\end{axiom}

\begin{axiom}

λ:=                      _
     (    I ΩX     )  /  _
     (     Y I     );

changed:
-Frobenius Condition (fork)
-\begin{axiom}
-
-H := Y / λ;
-
-test
-     (   λ I   )  /
-     (  I Y    )  =  H
-
-test
-     (   I λ   )  /
-     (    Y I  )  =  H
-
-\end{axiom}
-
-The Cartan-Killing form makes H of the Frobenius condition idempotent
-\begin{axiom}
-test( H=H/H )
-\end{axiom}
-
-Handle
-\begin{axiom}
-
-Φ := λ / Y;
-
-\end{axiom}
-
-The Cartan-Killing form makes Φ of the identity
-\begin{axiom}
-test( Φ=I )
-\end{axiom}
-
-Definition 5
-
-Unit
-\begin{axiom}
-
-e:=𝐞.1
Cartan-Killing co-multiplication
\begin{axiom}
eval(λ,ck)
\end{axiom}

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

\begin{axiom}


changed:
-Co-unit
-\begin{axiom}
-
-d:=
-    (    e I   ) /
-          Ũ
-test
-        Y     /
-        d     =  Ũ
-
-\end{axiom}
-
-Figure 12
-
-\begin{axiom}
-
-ΩXΩ:= ΩX * Ω;
-YXY:= Y * XY;
-arity(ΩXΩ)
-φφ := ΩXΩ / (I X I ) / YXY;
-
-φφ1:=map((x:ℚ):ℚ+->numer x,φφ)
-φφ2:=denom(ravel(φφ).1)
-test(φφ=(1/φφ2)*φφ1)
-\end{axiom}
-For Cartan-Killing this is just the co-scalar
-\begin{axiom}
-test(φφ=Ω)
-test((e,e)/H=Ω)
-\end{axiom}
-
-Bi-algebra conditions
-\begin{axiom}
-ΦΦ:=          _
-  (  λ λ  ) / _
-  ( I I X ) / _
-  ( I X I ) / _
-  (  YXY  ) ;
-test((e,e)/ΦΦ=φφ)
-test(ΦΦ=H/H)
-test(ΦΦ/(d,d)=Ũ)
-test(H/(d,d)=Ũ)
-\end{axiom}
-

The Pauli Algebra Cl(3) Is Frobenius In Many Ways
Linear operators over a 8-dimensional vector space representing Pauli algebra
Ref:


http://arxiv.org/abs/1103.5113  $S_3$-permuted Frobenius Algebras
  Zbigniew Oziewicz (UNAM), Gregory Peter Wene (UTSA)

http://mat.uab.es/~kock/TQFT.html  Frobenius algebras and 2D topological quantum field theories
  Joachim Kock

http://en.wikipedia.org/wiki/Frobenius_algebra
http://en.wikipedia.org/wiki/Pauli_matrices
http://en.wikipedia.org/wiki/Clifford_algebra


We need the Axiom LinearOperator library.
\begin{axiom}
)library CARTEN ARITY CMONAL CPROP CLOP CALEY
\end{axiom}
Use the following macros for convenient notation
\begin{axiom}
-- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])
-- list
macro Ξ(f,i,n)==[f for i in n]
-- subscript and superscripts
macro sb == subscript
macro sp == superscript
\end{axiom}
𝐋 is the domain of 8-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients.
\begin{axiom}
dim:=8
macro ℒ == List
macro ℂ == CaleyDickson
macro ℚ == Expression Integer
𝐋 := ClosedLinearOperator(OVAR ['1,'i,'j,'k,'ij,'ik,'jk,'ijk], ℚ)
𝐞:ℒ 𝐋      := basisOut()
𝐝:ℒ 𝐋      := basisIn()
I:𝐋:=[1]   -- identity for composition
X:𝐋:=[2,1] -- twist
V:𝐋:=ev(1) -- evaluation
Λ:𝐋:=co(1) -- co-evaluation
equate(eq)==map((x,y)+->(x=y),ravel lhs eq, ravel rhs eq);
\end{axiom}
Now generate structure constants for Pauli 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.
The Pauli Algebra as Cl(3)
Basis: Each B.i is a Clifford number
\begin{axiom}
q0:=sp('i,[2])
q1:=sp('j,[2])
q2:=sp('k,[2])
QQ:=CliffordAlgebra(3,ℚ,matrix [[q0,0,0],[0,q1,0],[0,0,q2]])
B:ℒ QQ := [monomial(1,[]),monomial(1,[1]),monomial(1,[2]),monomial(1,[3]),monomial(1,[1,2]),monomial(1,[1,3]),monomial(1,[2,3]),monomial(1,[1,2,3])]
M:Matrix QQ := matrix Ξ(Ξ(B.iB.j, i,1..dim), j,1..dim)
S(y) == map(x +-> coefficient(recip(y)x,[]),M)
ѕ :=map(S,B)::ℒ ℒ ℒ ℚ
-- 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)
matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y, i,1..dim), j,1..dim)
XY := X/Y;
\end{axiom}
Units
\begin{axiom}
e:=𝐞.1; i:=𝐞.2; j:=𝐞.3; k:=𝐞.4; ij:=𝐞.5; ik:=𝐞.6; jk:=𝐞.7; ijk:=𝐞.8;
\end{axiom}
Multiplication of arbitrary quaternions $a$ and $b$
\begin{axiom}
a:=Σ(sb('a,[i])*𝐞.i, i,1..dim)
b:=Σ(sb('b,[i])*𝐞.i, i,1..dim)
(a*b)/Y
\end{axiom}
Multiplication is Associative
\begin{axiom}
test(
  ( I Y ) / 
  (  Y  ) = 
  ( Y I ) / _
  (  Y  ) )
\end{axiom}
A scalar product is denoted by the (2,0)-tensor
$U = \{ u_{ij} \}$
\begin{axiom}
U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)
\end{axiom}
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:
$$
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$$
  \begin{equation}
\label{eq1}
  \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}
  \end{equation}
  (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.
\begin{axiom}
ω:𝐋 :=                 
     (    Y I    )  /  
           U        -  
     (    I Y    )  /  
           U;
\end{axiom}
Definition 2
  An algebra with a non-degenerate associative scalar product
  is called a [Frobenius Algebra]?.
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 := 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
Frobenius Form (co-unit)
\begin{axiom}
d:=ε1*𝐝.1+εi*𝐝.2+εj*𝐝.3+εk*𝐝.4+εij*𝐝.5+εik*𝐝.6+εjk*𝐝.7+εijk*𝐝.8
Ų:= Y/d
\end{axiom}
In general the pairing is not symmetric!
\begin{axiom}
u1:=matrix Ξ(Ξ(retract((𝐞.i 𝐞.j)/Ų), i,1..dim), j,1..dim)
--eigenvectors(u1::Matrix FRAC POLY INT)
\end{axiom}
The scalar product must be non-degenerate:
\begin{axiom}
--Ů:=determinant u1
--factor(numer Ů)/factor(denom Ů)
1
\end{axiom}
Cartan-Killing is a special case
\begin{axiom}
ck:=solve(equate(Ũ=Ų),[ε1,εi,εj,εk,εij,εik,εjk,εijk]?).1
\end{axiom}
Frobenius scalar product of "vector" quaternions $a$ and $b$
\begin{axiom}
a:=sb('a,[1]?)*i+sb('a,[2]?)*j+sb('a,[3]?)*k
b:=sb('b,[1]?)*i+sb('b,[2]?)*j+sb('b,[3]?)*k
(a,a)/Ų
(b,b)/Ų
(a,b)/Ų
\end{axiom}
Definition 3
  Co-scalar product
Solve the [Snake Relation]? as a system of linear equations.
\begin{axiom}
mU:=inverse matrix Ξ(Ξ(retract((𝐞.i*𝐞.j)/Ų), i,1..dim), j,1..dim);
Ω:=Σ(Σ(mU(i,j)*(𝐞.i*𝐞.j), i,1..dim), j,1..dim);
ΩX:=Ω/X;
--matrix Ξ(Ξ(Ω/(𝐝.i*𝐝.j), i,1..dim), j,1..dim)
\end{axiom}
The common demoninator is $1/\sqrt{\mathring{U}}$
\begin{axiom}
--squareFreePart factor denom Ů / squareFreePart factor numer Ů
matrix Ξ(Ξ(numer retract(Ω/(𝐝.i*𝐝.j)), i,1..dim), j,1..dim)
\end{axiom}
Check "dimension" and the snake relations.
\begin{axiom}
O:𝐋:= Ω / Ų
test
    (    I ΩX     )  /
    (     Ų I     )  =  I
test
    (     ΩX I    )  /
    (    I Ų      )  =  I
\end{axiom}
Cartan-Killing co-scalar
\begin{axiom}
eval(Ω,ck)
\end{axiom}
Definition 4
  Co-algebra
Compute the "three-point" function and use it to define co-multiplication.
\begin{axiom}
W:= (Y I) / Ų;
  (  ΩX I ΩX  ) /
  (  I  W  I  )
\end{axiom}
\begin{axiom}
λ:=                      _
     (    I ΩX     )  /  _
     (     Y I     );
test
     (     ΩX I    )  /
     (    I  Y     )  =  λ
\end{axiom}
Cartan-Killing co-multiplication
\begin{axiom}
eval(λ,ck)
\end{axiom}
Co-associativity
\begin{axiom}
test(
  (  λ  ) / _
  ( I λ ) = _
  (  λ  ) / _
  ( λ I ) )
\end{axiom}
\begin{axiom}
test
         e     /
         λ     =    ΩX
\end{axiom}


    Some or all expressions may not have rendered properly,
    because Axiom returned the following error:
Error: export AXIOM=/usr/local/lib/fricas/target/x86_64-unknown-linux; export ALDORROOT=/usr/local/aldor/linux/1.1.0; export PATH=$ALDORROOT/bin:$PATH; export HOME=/var/zope2/var/LatexWiki; ulimit -t 600; export LD_LIBRARY_PATH=/usr/local/lib/fricas/target/x86_64-unknown-linux/lib; LANG=en_US.UTF-8 $AXIOM/bin/AXIOMsys < /var/zope2/var/LatexWiki/7197492308318484242-25px.axm
Killed
Checking for foreign routines
AXIOM="/usr/local/lib/fricas/target/x86_64-unknown-linux"
spad-lib="/usr/local/lib/fricas/target/x86_64-unknown-linux/lib/libspad.so"
foreign routines found
openServer result -2
                 FriCAS (AXIOM fork) Computer Algebra System 
                         Version: FriCAS 2010-12-08
                Timestamp: Tuesday April 5, 2011 at 13:07:45 
-----------------------------------------------------------------------------
   Issue )copyright to view copyright notices.
   Issue )summary for a summary of useful system commands.
   Issue )quit to leave FriCAS and return to shell.
-----------------------------------------------------------------------------
(1) -> (1) -> (1) -> (1) -> (1) -> )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
(1) -> -- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])

                                                                   Type: Void
list
macro Ξ(f,i,n)==[f for i in n]
                                                                   Type: Void
subscript and superscripts
macro sb == subscript

                                                                   Type: Void
macro sp == superscript
                                                                   Type: Void
(5) -> dim:=8

$$
8 
\leqno(5)
$$
                                                        Type: PositiveInteger
macro ℒ == List
                                                                   Type: Void
macro ℂ == CaleyDickson
                                                                   Type: Void
macro ℚ == Expression Integer
                                                                   Type: Void
𝐋 := ClosedLinearOperator(OVAR ['1,'i,'j,'k,'ij,'ik,'jk,'ijk], ℚ)


$$
ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
\leqno(9)
$$
                                                                   Type: Type
𝐞:ℒ 𝐋      := basisOut()

$$
\left[
{| \sb {{ \  1}}}, \: {| \sb {{ \  i}}}, \: {| \sb {{ \  j}}}, \: {| \sb {{ \  
k}}}, \: {| \sb {{ \  ij}}}, \: {| \sb {{ \  ik}}}, \: {| \sb {{ \  jk}}}, \: 
{| \sb {{ \  ijk}}} 
\right]
\leqno(10)
$$
Type: List(ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)))
𝐝:ℒ 𝐋      := basisIn()
$$
\left[
{| \sp {{ \  1}}}, \: {| \sp {{ \  i}}}, \: {| \sp {{ \  j}}}, \: {| \sp {{ \  
k}}}, \: {| \sp {{ \  ij}}}, \: {| \sp {{ \  ik}}}, \: {| \sp {{ \  jk}}}, \: 
{| \sp {{ \  ijk}}} 
\right]
\leqno(11)
$$
Type: List(ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)))
I:𝐋:=[1]   -- identity for composition
$$
{| \sb {{ \  1}} \sp {{ \  1}}}+{| \sb {{ \  i}} \sp {{ \  i}}}+{| \sb {{ \  
j}} \sp {{ \  j}}}+{| \sb {{ \  k}} \sp {{ \  k}}}+{| \sb {{ \  ij}} \sp {{ \  
ij}}}+{| \sb {{ \  ik}} \sp {{ \  ik}}}+{| \sb {{ \  jk}} \sp {{ \  jk}}}+{| 
\sb {{ \  ijk}} \sp {{ \  ijk}}} 
\leqno(12)
$$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
X:𝐋:=[2,1] -- twist
$$
{| \sb {{ \  1 \  1}} \sp {{ \  1 \  1}}}+{| \sb {{ \  i \  1}} \sp {{ \  1 \  
i}}}+{| \sb {{ \  j \  1}} \sp {{ \  1 \  j}}}+{| \sb {{ \  k \  1}} \sp {{ \  
1 \  k}}}+{| \sb {{ \  ij \  1}} \sp {{ \  1 \  ij}}}+{| \sb {{ \  ik \  1}} 
\sp {{ \  1 \  ik}}}+{| \sb {{ \  jk \  1}} \sp {{ \  1 \  jk}}}+{| \sb {{ \  
ijk \  1}} \sp {{ \  1 \  ijk}}}+{| \sb {{ \  1 \  i}} \sp {{ \  i \  1}}}+{| 
\sb {{ \  i \  i}} \sp {{ \  i \  i}}}+{| \sb {{ \  j \  i}} \sp {{ \  i \  
j}}}+{| \sb {{ \  k \  i}} \sp {{ \  i \  k}}}+{| \sb {{ \  ij \  i}} \sp {{ 
\  i \  ij}}}+{| \sb {{ \  ik \  i}} \sp {{ \  i \  ik}}}+{| \sb {{ \  jk \  
i}} \sp {{ \  i \  jk}}}+{| \sb {{ \  ijk \  i}} \sp {{ \  i \  ijk}}}+{| \sb 
{{ \  1 \  j}} \sp {{ \  j \  1}}}+{| \sb {{ \  i \  j}} \sp {{ \  j \  
i}}}+{| \sb {{ \  j \  j}} \sp {{ \  j \  j}}}+{| \sb {{ \  k \  j}} \sp {{ \  
j \  k}}}+{| \sb {{ \  ij \  j}} \sp {{ \  j \  ij}}}+{| \sb {{ \  ik \  j}} 
\sp {{ \  j \  ik}}}+{| \sb {{ \  jk \  j}} \sp {{ \  j \  jk}}}+{| \sb {{ \  
ijk \  j}} \sp {{ \  j \  ijk}}}+{| \sb {{ \  1 \  k}} \sp {{ \  k \  1}}}+{| 
\sb {{ \  i \  k}} \sp {{ \  k \  i}}}+{| \sb {{ \  j \  k}} \sp {{ \  k \  
j}}}+{| \sb {{ \  k \  k}} \sp {{ \  k \  k}}}+{| \sb {{ \  ij \  k}} \sp {{ 
\  k \  ij}}}+{| \sb {{ \  ik \  k}} \sp {{ \  k \  ik}}}+{| \sb {{ \  jk \  
k}} \sp {{ \  k \  jk}}}+{| \sb {{ \  ijk \  k}} \sp {{ \  k \  ijk}}}+{| \sb 
{{ \  1 \  ij}} \sp {{ \  ij \  1}}}+{| \sb {{ \  i \  ij}} \sp {{ \  ij \  
i}}}+{| \sb {{ \  j \  ij}} \sp {{ \  ij \  j}}}+{| \sb {{ \  k \  ij}} \sp 
{{ \  ij \  k}}}+{| \sb {{ \  ij \  ij}} \sp {{ \  ij \  ij}}}+{| \sb {{ \  
ik \  ij}} \sp {{ \  ij \  ik}}}+{| \sb {{ \  jk \  ij}} \sp {{ \  ij \  
jk}}}+{| \sb {{ \  ijk \  ij}} \sp {{ \  ij \  ijk}}}+{| \sb {{ \  1 \  ik}} 
\sp {{ \  ik \  1}}}+{| \sb {{ \  i \  ik}} \sp {{ \  ik \  i}}}+{| \sb {{ \  
j \  ik}} \sp {{ \  ik \  j}}}+{| \sb {{ \  k \  ik}} \sp {{ \  ik \  k}}}+{| 
\sb {{ \  ij \  ik}} \sp {{ \  ik \  ij}}}+{| \sb {{ \  ik \  ik}} \sp {{ \  
ik \  ik}}}+{| \sb {{ \  jk \  ik}} \sp {{ \  ik \  jk}}}+{| \sb {{ \  ijk \  
ik}} \sp {{ \  ik \  ijk}}}+{| \sb {{ \  1 \  jk}} \sp {{ \  jk \  1}}}+{| 
\sb {{ \  i \  jk}} \sp {{ \  jk \  i}}}+{| \sb {{ \  j \  jk}} \sp {{ \  jk 
\  j}}}+{| \sb {{ \  k \  jk}} \sp {{ \  jk \  k}}}+{| \sb {{ \  ij \  jk}} 
\sp {{ \  jk \  ij}}}+{| \sb {{ \  ik \  jk}} \sp {{ \  jk \  ik}}}+{| \sb {{ 
\  jk \  jk}} \sp {{ \  jk \  jk}}}+{| \sb {{ \  ijk \  jk}} \sp {{ \  jk \  
ijk}}}+{| \sb {{ \  1 \  ijk}} \sp {{ \  ijk \  1}}}+{| \sb {{ \  i \  ijk}} 
\sp {{ \  ijk \  i}}}+{| \sb {{ \  j \  ijk}} \sp {{ \  ijk \  j}}}+{| \sb {{ 
\  k \  ijk}} \sp {{ \  ijk \  k}}}+{| \sb {{ \  ij \  ijk}} \sp {{ \  ijk \  
ij}}}+{| \sb {{ \  ik \  ijk}} \sp {{ \  ijk \  ik}}}+{| \sb {{ \  jk \  
ijk}} \sp {{ \  ijk \  jk}}}+{| \sb {{ \  ijk \  ijk}} \sp {{ \  ijk \  
ijk}}} 
\leqno(13)
$$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
V:𝐋:=ev(1) -- evaluation
$$
{| \sp {{ \  1 \  1}}}+{| \sp {{ \  i \  i}}}+{| \sp {{ \  j \  j}}}+{| \sp 
{{ \  k \  k}}}+{| \sp {{ \  ij \  ij}}}+{| \sp {{ \  ik \  ik}}}+{| \sp {{ \  
jk \  jk}}}+{| \sp {{ \  ijk \  ijk}}} 
\leqno(14)
$$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
Λ:𝐋:=co(1) -- co-evaluation
$$
{| \sb {{ \  1 \  1}}}+{| \sb {{ \  i \  i}}}+{| \sb {{ \  j \  j}}}+{| \sb 
{{ \  k \  k}}}+{| \sb {{ \  ij \  ij}}}+{| \sb {{ \  ik \  ik}}}+{| \sb {{ \  
jk \  jk}}}+{| \sb {{ \  ijk \  ijk}}} 
\leqno(15)
$$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
equate(eq)==map((x,y)+->(x=y),ravel lhs eq, ravel rhs eq);
                                                                   Type: Void
(17) -> q0:=sp('i,[2])

$$
i \sp {2} 
\leqno(17)
$$
                                                                 Type: Symbol
q1:=sp('j,[2])

$$
j \sp {2} 
\leqno(18)
$$
                                                                 Type: Symbol
q2:=sp('k,[2])

$$
k \sp {2} 
\leqno(19)
$$
                                                                 Type: Symbol
QQ:=CliffordAlgebra(3,ℚ,matrix [[q0,0,0],[0,q1,0],[0,0,q2]])

$$
CliffordAlgebra(3,Expression(Integer),[[*001i(2),0,0],[0,*001j(2),0],[0,0,*001k(2)]])
\leqno(20)
$$
                                                                   Type: Type
B:ℒ QQ := [monomial(1,[]),monomial(1,[1]),monomial(1,[2]),monomial(1,[3]),monomial(1,[1,2]),monomial(1,[1,3]),monomial(1,[2,3]),monomial(1,[1,2,3])]

$$
\left[
1, \: {e \sb {1}}, \: {e \sb {2}}, \: {e \sb {3}}, \: {{e \sb {1}} \  {e \sb 
{2}}}, \: {{e \sb {1}} \  {e \sb {3}}}, \: {{e \sb {2}} \  {e \sb {3}}}, \: 
{{e \sb {1}} \  {e \sb {2}} \  {e \sb {3}}} 
\right]
\leqno(21)
$$
Type: List(CliffordAlgebra(3,Expression(Integer),[[*001i(2),0,0],[0,*001j(2),0],[0,0,*001k(2)]]))
M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j, i,1..dim), j,1..dim)
$$
\left[
\begin{array}{cccccccc}
1 & {e \sb {1}} & {e \sb {2}} & {e \sb {3}} & {{e \sb {1}} \  {e \sb {2}}} & 
{{e \sb {1}} \  {e \sb {3}}} & {{e \sb {2}} \  {e \sb {3}}} & {{e \sb {1}} \  
{e \sb {2}} \  {e \sb {3}}} \ 
{e \sb {1}} & {i \sp {2}} & -{{e \sb {1}} \  {e \sb {2}}} & -{{e \sb {1}} \  
{e \sb {3}}} & -{{i \sp {2}} \  {e \sb {2}}} & -{{i \sp {2}} \  {e \sb {3}}} 
& {{e \sb {1}} \  {e \sb {2}} \  {e \sb {3}}} & {{i \sp {2}} \  {e \sb {2}} \  
{e \sb {3}}} \ 
{e \sb {2}} & {{e \sb {1}} \  {e \sb {2}}} & {j \sp {2}} & -{{e \sb {2}} \  
{e \sb {3}}} & {{j \sp {2}} \  {e \sb {1}}} & -{{e \sb {1}} \  {e \sb {2}} \  
{e \sb {3}}} & -{{j \sp {2}} \  {e \sb {3}}} & -{{j \sp {2}} \  {e \sb {1}} \  
{e \sb {3}}} \ 
{e \sb {3}} & {{e \sb {1}} \  {e \sb {3}}} & {{e \sb {2}} \  {e \sb {3}}} & 
{k \sp {2}} & {{e \sb {1}} \  {e \sb {2}} \  {e \sb {3}}} & {{k \sp {2}} \  
{e \sb {1}}} & {{k \sp {2}} \  {e \sb {2}}} & {{k \sp {2}} \  {e \sb {1}} \  
{e \sb {2}}} \ 
{{e \sb {1}} \  {e \sb {2}}} & {{i \sp {2}} \  {e \sb {2}}} & -{{j \sp {2}} \  
{e \sb {1}}} & {{e \sb {1}} \  {e \sb {2}} \  {e \sb {3}}} & -{{i \sp {2}} \  
{j \sp {2}}} & {{i \sp {2}} \  {e \sb {2}} \  {e \sb {3}}} & -{{j \sp {2}} \  
{e \sb {1}} \  {e \sb {3}}} & -{{i \sp {2}} \  {j \sp {2}} \  {e \sb {3}}} \ 
{{e \sb {1}} \  {e \sb {3}}} & {{i \sp {2}} \  {e \sb {3}}} & -{{e \sb {1}} \  
{e \sb {2}} \  {e \sb {3}}} & -{{k \sp {2}} \  {e \sb {1}}} & -{{i \sp {2}} \  
{e \sb {2}} \  {e \sb {3}}} & -{{i \sp {2}} \  {k \sp {2}}} & {{k \sp {2}} \  
{e \sb {1}} \  {e \sb {2}}} & {{i \sp {2}} \  {k \sp {2}} \  {e \sb {2}}} \ 
{{e \sb {2}} \  {e \sb {3}}} & {{e \sb {1}} \  {e \sb {2}} \  {e \sb {3}}} & 
{{j \sp {2}} \  {e \sb {3}}} & -{{k \sp {2}} \  {e \sb {2}}} & {{j \sp {2}} \  
{e \sb {1}} \  {e \sb {3}}} & -{{k \sp {2}} \  {e \sb {1}} \  {e \sb {2}}} & 
-{{j \sp {2}} \  {k \sp {2}}} & -{{j \sp {2}} \  {k \sp {2}} \  {e \sb {1}}} \ 
{{e \sb {1}} \  {e \sb {2}} \  {e \sb {3}}} & {{i \sp {2}} \  {e \sb {2}} \  
{e \sb {3}}} & -{{j \sp {2}} \  {e \sb {1}} \  {e \sb {3}}} & {{k \sp {2}} \  
{e \sb {1}} \  {e \sb {2}}} & -{{i \sp {2}} \  {j \sp {2}} \  {e \sb {3}}} & 
{{i \sp {2}} \  {k \sp {2}} \  {e \sb {2}}} & -{{j \sp {2}} \  {k \sp {2}} \  
{e \sb {1}}} & -{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}}} 
\end{array}
\right]
\leqno(22)
$$
Type: Matrix(CliffordAlgebra(3,Expression(Integer),[[*001i(2),0,0],[0,*001j(2),0],[0,0,001k(2)]]))
S(y) == map(x +-> coefficient(recip(y)x,[]),M)
                                                                   Type: Void
ѕ :=map(S,B)::ℒ ℒ ℒ ℚ
   Compiling function S with type CliffordAlgebra(3,Expression(Integer)
      ,[[*001i(2),0,0],[0,*001j(2),0],[0,0,*001k(2)]]) -> Matrix(
      Expression(Integer)) 

$$
\left[
{\left[ {\left[ 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: {i \sp {2}}, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: {j \sp {2}}, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: {k \sp {2}}, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: -{{i \sp {2}} \  {j \sp {2}}}, \: 0, \: 0, 
\: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: -{{i \sp {2}} \  {k \sp {2}}}, \: 0, 
\: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{{j \sp {2}} \  {k \sp {2}}}, 
\: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{{i \sp {2}} \  {j \sp 
{2}} \  {k \sp {2}}} 
\right]}
\right]},
\: {\left[ {\left[ 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: {j \sp {2}}, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: {k \sp {2}}, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: -{j \sp {2}}, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: -{k \sp {2}}, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{{j \sp {2}} \  {k \sp 
{2}}} 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{{j \sp {2}} \  {k \sp {2}}}, 
\: 0 
\right]}
\right]},
\: {\left[ {\left[ 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: -{i \sp {2}}, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: {k \sp {2}}, \: 0 
\right]},
\: {\left[ 0, \: {i \sp {2}}, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: {{i \sp {2}} \  {k \sp 
{2}}} 
\right]},
\: {\left[ 0, \: 0, \: 0, \: -{k \sp {2}}, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: {{i \sp {2}} \  {k \sp {2}}}, \: 0, 
\: 0 
\right]}
\right]},
\: {\left[ {\left[ 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: -{i \sp {2}}, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{j \sp {2}}, \: 0 
\right]},
\: {\left[ 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{{i \sp {2}} \  {j \sp 
{2}}} 
\right]},
\: {\left[ 0, \: {i \sp {2}}, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: {j \sp {2}}, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: -{{i \sp {2}} \  {j \sp {2}}}, \: 0, \: 0, 
\: 0 
\right]}
\right]},
\: {\left[ {\left[ 0, \: 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: -1, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: {k \sp {2}} 
\right]},
\: {\left[ 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: {k \sp {2}}, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: -{k \sp {2}}, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: {k \sp {2}}, \: 0, \: 0, \: 0, \: 0 
\right]}
\right]},
\: {\left[ {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 1, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: -1, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{j \sp {2}} 
\right]},
\: {\left[ 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{j \sp {2}}, \: 0 
\right]},
\: {\left[ 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: {j \sp {2}}, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: -{j \sp {2}}, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]}
\right]},
\: {\left[ {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 1, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: {i \sp {2}} 
\right]},
\: {\left[ 0, \: 0, \: 0, \: -1, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: {i \sp {2}}, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: -{i \sp {2}}, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: {i \sp {2}}, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]}
\right]},
\: {\left[ {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 1 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 1, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 0, \: -1, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 0, \: -1, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]},
\: {\left[ 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 
\right]}
\right]}
\right]
\leqno(24)
$$

                                  Type: List(List(List(Expression(Integer))))
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)


$$
{| \sb {{ \  1}} \sp {{ \  1 \  1}}}+{| \sb {{ \  i}} \sp {{ \  1 \  i}}}+{| 
\sb {{ \  j}} \sp {{ \  1 \  j}}}+{| \sb {{ \  k}} \sp {{ \  1 \  k}}}+{| \sb 
{{ \  ij}} \sp {{ \  1 \  ij}}}+{| \sb {{ \  ik}} \sp {{ \  1 \  ik}}}+{| \sb 
{{ \  jk}} \sp {{ \  1 \  jk}}}+{| \sb {{ \  ijk}} \sp {{ \  1 \  ijk}}}+{| 
\sb {{ \  i}} \sp {{ \  i \  1}}}+{{i \sp {2}} \  {| \sb {{ \  1}} \sp {{ \  
i \  i}}}}+{| \sb {{ \  ij}} \sp {{ \  i \  j}}}+{| \sb {{ \  ik}} \sp {{ \  
i \  k}}}+{{i \sp {2}} \  {| \sb {{ \  j}} \sp {{ \  i \  ij}}}}+{{i \sp {2}} 
\  {| \sb {{ \  k}} \sp {{ \  i \  ik}}}}+{| \sb {{ \  ijk}} \sp {{ \  i \  
jk}}}+{{i \sp {2}} \  {| \sb {{ \  jk}} \sp {{ \  i \  ijk}}}}+{| \sb {{ \  
j}} \sp {{ \  j \  1}}} -{| \sb {{ \  ij}} \sp {{ \  j \  i}}}+{{j \sp {2}} \  
{| \sb {{ \  1}} \sp {{ \  j \  j}}}}+{| \sb {{ \  jk}} \sp {{ \  j \  k}}} 
-{{j \sp {2}} \  {| \sb {{ \  i}} \sp {{ \  j \  ij}}}} -{| \sb {{ \  ijk}} 
\sp {{ \  j \  ik}}}+{{j \sp {2}} \  {| \sb {{ \  k}} \sp {{ \  j \  jk}}}} 
-{{j \sp {2}} \  {| \sb {{ \  ik}} \sp {{ \  j \  ijk}}}}+{| \sb {{ \  k}} 
\sp {{ \  k \  1}}} -{| \sb {{ \  ik}} \sp {{ \  k \  i}}} -{| \sb {{ \  jk}} 
\sp {{ \  k \  j}}}+{{k \sp {2}} \  {| \sb {{ \  1}} \sp {{ \  k \  k}}}}+{| 
\sb {{ \  ijk}} \sp {{ \  k \  ij}}} -{{k \sp {2}} \  {| \sb {{ \  i}} \sp {{ 
\  k \  ik}}}} -{{k \sp {2}} \  {| \sb {{ \  j}} \sp {{ \  k \  jk}}}}+{{k 
\sp {2}} \  {| \sb {{ \  ij}} \sp {{ \  k \  ijk}}}}+{| \sb {{ \  ij}} \sp {{ 
\  ij \  1}}} -{{i \sp {2}} \  {| \sb {{ \  j}} \sp {{ \  ij \  i}}}}+{{j \sp 
{2}} \  {| \sb {{ \  i}} \sp {{ \  ij \  j}}}}+{| \sb {{ \  ijk}} \sp {{ \  
ij \  k}}} -{{i \sp {2}} \  {j \sp {2}} \  {| \sb {{ \  1}} \sp {{ \  ij \  
ij}}}} -{{i \sp {2}} \  {| \sb {{ \  jk}} \sp {{ \  ij \  ik}}}}+{{j \sp {2}} 
\  {| \sb {{ \  ik}} \sp {{ \  ij \  jk}}}} -{{i \sp {2}} \  {j \sp {2}} \  
{| \sb {{ \  k}} \sp {{ \  ij \  ijk}}}}+{| \sb {{ \  ik}} \sp {{ \  ik \  
1}}} -{{i \sp {2}} \  {| \sb {{ \  k}} \sp {{ \  ik \  i}}}} -{| \sb {{ \  
ijk}} \sp {{ \  ik \  j}}}+{{k \sp {2}} \  {| \sb {{ \  i}} \sp {{ \  ik \  
k}}}}+{{i \sp {2}} \  {| \sb {{ \  jk}} \sp {{ \  ik \  ij}}}} -{{i \sp {2}} 
\  {k \sp {2}} \  {| \sb {{ \  1}} \sp {{ \  ik \  ik}}}} -{{k \sp {2}} \  {| 
\sb {{ \  ij}} \sp {{ \  ik \  jk}}}}+{{i \sp {2}} \  {k \sp {2}} \  {| \sb 
{{ \  j}} \sp {{ \  ik \  ijk}}}}+{| \sb {{ \  jk}} \sp {{ \  jk \  1}}}+{| 
\sb {{ \  ijk}} \sp {{ \  jk \  i}}} -{{j \sp {2}} \  {| \sb {{ \  k}} \sp {{ 
\  jk \  j}}}}+{{k \sp {2}} \  {| \sb {{ \  j}} \sp {{ \  jk \  k}}}} -{{j 
\sp {2}} \  {| \sb {{ \  ik}} \sp {{ \  jk \  ij}}}}+{{k \sp {2}} \  {| \sb 
{{ \  ij}} \sp {{ \  jk \  ik}}}} -{{j \sp {2}} \  {k \sp {2}} \  {| \sb {{ \  
1}} \sp {{ \  jk \  jk}}}} -{{j \sp {2}} \  {k \sp {2}} \  {| \sb {{ \  i}} 
\sp {{ \  jk \  ijk}}}}+{| \sb {{ \  ijk}} \sp {{ \  ijk \  1}}}+{{i \sp {2}} 
\  {| \sb {{ \  jk}} \sp {{ \  ijk \  i}}}} -{{j \sp {2}} \  {| \sb {{ \  
ik}} \sp {{ \  ijk \  j}}}}+{{k \sp {2}} \  {| \sb {{ \  ij}} \sp {{ \  ijk \  
k}}}} -{{i \sp {2}} \  {j \sp {2}} \  {| \sb {{ \  k}} \sp {{ \  ijk \  
ij}}}}+{{i \sp {2}} \  {k \sp {2}} \  {| \sb {{ \  j}} \sp {{ \  ijk \  
ik}}}} -{{j \sp {2}} \  {k \sp {2}} \  {| \sb {{ \  i}} \sp {{ \  ijk \  
jk}}}} -{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {| \sb {{ \  1}} \sp {{ 
\  ijk \  ijk}}}} 
\leqno(25)
$$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y, i,1..dim), j,1..dim)
$$
\left[
\begin{array}{cccccccc}
{| \sb {{ \  1}}} & {| \sb {{ \  i}}} & {| \sb {{ \  j}}} & {| \sb {{ \  k}}} 
& {| \sb {{ \  ij}}} & {| \sb {{ \  ik}}} & {| \sb {{ \  jk}}} & {| \sb {{ \  
ijk}}} \ 
{| \sb {{ \  i}}} & {{i \sp {2}} \  {| \sb {{ \  1}}}} & -{| \sb {{ \  ij}}} 
& -{| \sb {{ \  ik}}} & -{{i \sp {2}} \  {| \sb {{ \  j}}}} & -{{i \sp {2}} \  
{| \sb {{ \  k}}}} & {| \sb {{ \  ijk}}} & {{i \sp {2}} \  {| \sb {{ \  
jk}}}} \ 
{| \sb {{ \  j}}} & {| \sb {{ \  ij}}} & {{j \sp {2}} \  {| \sb {{ \  1}}}} & 
-{| \sb {{ \  jk}}} & {{j \sp {2}} \  {| \sb {{ \  i}}}} & -{| \sb {{ \  
ijk}}} & -{{j \sp {2}} \  {| \sb {{ \  k}}}} & -{{j \sp {2}} \  {| \sb {{ \  
ik}}}} \ 
{| \sb {{ \  k}}} & {| \sb {{ \  ik}}} & {| \sb {{ \  jk}}} & {{k \sp {2}} \  
{| \sb {{ \  1}}}} & {| \sb {{ \  ijk}}} & {{k \sp {2}} \  {| \sb {{ \  i}}}} 
& {{k \sp {2}} \  {| \sb {{ \  j}}}} & {{k \sp {2}} \  {| \sb {{ \  ij}}}} \ 
{| \sb {{ \  ij}}} & {{i \sp {2}} \  {| \sb {{ \  j}}}} & -{{j \sp {2}} \  {| 
\sb {{ \  i}}}} & {| \sb {{ \  ijk}}} & -{{i \sp {2}} \  {j \sp {2}} \  {| 
\sb {{ \  1}}}} & {{i \sp {2}} \  {| \sb {{ \  jk}}}} & -{{j \sp {2}} \  {| 
\sb {{ \  ik}}}} & -{{i \sp {2}} \  {j \sp {2}} \  {| \sb {{ \  k}}}} \ 
{| \sb {{ \  ik}}} & {{i \sp {2}} \  {| \sb {{ \  k}}}} & -{| \sb {{ \  
ijk}}} & -{{k \sp {2}} \  {| \sb {{ \  i}}}} & -{{i \sp {2}} \  {| \sb {{ \  
jk}}}} & -{{i \sp {2}} \  {k \sp {2}} \  {| \sb {{ \  1}}}} & {{k \sp {2}} \  
{| \sb {{ \  ij}}}} & {{i \sp {2}} \  {k \sp {2}} \  {| \sb {{ \  j}}}} \ 
{| \sb {{ \  jk}}} & {| \sb {{ \  ijk}}} & {{j \sp {2}} \  {| \sb {{ \  k}}}} 
& -{{k \sp {2}} \  {| \sb {{ \  j}}}} & {{j \sp {2}} \  {| \sb {{ \  ik}}}} & 
-{{k \sp {2}} \  {| \sb {{ \  ij}}}} & -{{j \sp {2}} \  {k \sp {2}} \  {| \sb 
{{ \  1}}}} & -{{j \sp {2}} \  {k \sp {2}} \  {| \sb {{ \  i}}}} \ 
{| \sb {{ \  ijk}}} & {{i \sp {2}} \  {| \sb {{ \  jk}}}} & -{{j \sp {2}} \  
{| \sb {{ \  ik}}}} & {{k \sp {2}} \  {| \sb {{ \  ij}}}} & -{{i \sp {2}} \  
{j \sp {2}} \  {| \sb {{ \  k}}}} & {{i \sp {2}} \  {k \sp {2}} \  {| \sb {{ 
\  j}}}} & -{{j \sp {2}} \  {k \sp {2}} \  {| \sb {{ \  i}}}} & -{{i \sp {2}} 
\  {j \sp {2}} \  {k \sp {2}} \  {| \sb {{ \  1}}}} 
\end{array}
\right]
\leqno(26)
$$
Type: Matrix(ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)))
XY := X/Y;
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
(28) -> e:=𝐞.1; i:=𝐞.2; j:=𝐞.3; k:=𝐞.4; ij:=𝐞.5; ik:=𝐞.6; jk:=𝐞.7; ijk:=𝐞.8;
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
(29) -> a:=Σ(sb('a,[i])*𝐞.i, i,1..dim)
$$
{{a \sb {1}} \  {| \sb {{ \  1}}}}+{{a \sb {2}} \  {| \sb {{ \  i}}}}+{{a \sb 
{3}} \  {| \sb {{ \  j}}}}+{{a \sb {4}} \  {| \sb {{ \  k}}}}+{{a \sb {5}} \  
{| \sb {{ \  ij}}}}+{{a \sb {6}} \  {| \sb {{ \  ik}}}}+{{a \sb {7}} \  {| 
\sb {{ \  jk}}}}+{{a \sb {8}} \  {| \sb {{ \  ijk}}}} 
\leqno(29)
$$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
b:=Σ(sb('b,[i])*𝐞.i, i,1..dim)
$$
{{b \sb {1}} \  {| \sb {{ \  1}}}}+{{b \sb {2}} \  {| \sb {{ \  i}}}}+{{b \sb 
{3}} \  {| \sb {{ \  j}}}}+{{b \sb {4}} \  {| \sb {{ \  k}}}}+{{b \sb {5}} \  
{| \sb {{ \  ij}}}}+{{b \sb {6}} \  {| \sb {{ \  ik}}}}+{{b \sb {7}} \  {| 
\sb {{ \  jk}}}}+{{b \sb {8}} \  {| \sb {{ \  ijk}}}} 
\leqno(30)
$$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
(a*b)/Y
$$
{{\left( -{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {a \sb {8}} \  {b \sb 
{8}}} -{{j \sp {2}} \  {k \sp {2}} \  {a \sb {7}} \  {b \sb {7}}} -{{i \sp 
{2}} \  {k \sp {2}} \  {a \sb {6}} \  {b \sb {6}}} -{{i \sp {2}} \  {j \sp 
{2}} \  {a \sb {5}} \  {b \sb {5}}}+{{k \sp {2}} \  {a \sb {4}} \  {b \sb 
{4}}}+{{j \sp {2}} \  {a \sb {3}} \  {b \sb {3}}}+{{i \sp {2}} \  {a \sb {2}} 
\  {b \sb {2}}}+{{a \sb {1}} \  {b \sb {1}}} 
\right)}
\  {| \sb {{ \  1}}}}+{{\left( -{{j \sp {2}} \  {k \sp {2}} \  {a \sb {7}} \  
{b \sb {8}}} -{{j \sp {2}} \  {k \sp {2}} \  {a \sb {8}} \  {b \sb {7}}} -{{k 
\sp {2}} \  {a \sb {4}} \  {b \sb {6}}} -{{j \sp {2}} \  {a \sb {3}} \  {b 
\sb {5}}}+{{k \sp {2}} \  {a \sb {6}} \  {b \sb {4}}}+{{j \sp {2}} \  {a \sb 
{5}} \  {b \sb {3}}}+{{a \sb {1}} \  {b \sb {2}}}+{{a \sb {2}} \  {b \sb 
{1}}} 
\right)}
\  {| \sb {{ \  i}}}}+{{\left( {{i \sp {2}} \  {k \sp {2}} \  {a \sb {6}} \  
{b \sb {8}}} -{{k \sp {2}} \  {a \sb {4}} \  {b \sb {7}}}+{{i \sp {2}} \  {k 
\sp {2}} \  {a \sb {8}} \  {b \sb {6}}}+{{i \sp {2}} \  {a \sb {2}} \  {b \sb 
{5}}}+{{k \sp {2}} \  {a \sb {7}} \  {b \sb {4}}}+{{a \sb {1}} \  {b \sb 
{3}}} -{{i \sp {2}} \  {a \sb {5}} \  {b \sb {2}}}+{{a \sb {3}} \  {b \sb 
{1}}} 
\right)}
\  {| \sb {{ \  j}}}}+{{\left( -{{i \sp {2}} \  {j \sp {2}} \  {a \sb {5}} \  
{b \sb {8}}}+{{j \sp {2}} \  {a \sb {3}} \  {b \sb {7}}}+{{i \sp {2}} \  {a 
\sb {2}} \  {b \sb {6}}} -{{i \sp {2}} \  {j \sp {2}} \  {a \sb {8}} \  {b 
\sb {5}}}+{{a \sb {1}} \  {b \sb {4}}} -{{j \sp {2}} \  {a \sb {7}} \  {b \sb 
{3}}} -{{i \sp {2}} \  {a \sb {6}} \  {b \sb {2}}}+{{a \sb {4}} \  {b \sb 
{1}}} 
\right)}
\  {| \sb {{ \  k}}}}+{{\left( {{k \sp {2}} \  {a \sb {4}} \  {b \sb {8}}} 
-{{k \sp {2}} \  {a \sb {6}} \  {b \sb {7}}}+{{k \sp {2}} \  {a \sb {7}} \  
{b \sb {6}}}+{{a \sb {1}} \  {b \sb {5}}}+{{k \sp {2}} \  {a \sb {8}} \  {b 
\sb {4}}}+{{a \sb {2}} \  {b \sb {3}}} -{{a \sb {3}} \  {b \sb {2}}}+{{a \sb 
{5}} \  {b \sb {1}}} 
\right)}
\  {| \sb {{ \  ij}}}}+{{\left( -{{j \sp {2}} \  {a \sb {3}} \  {b \sb 
{8}}}+{{j \sp {2}} \  {a \sb {5}} \  {b \sb {7}}}+{{a \sb {1}} \  {b \sb 
{6}}} -{{j \sp {2}} \  {a \sb {7}} \  {b \sb {5}}}+{{a \sb {2}} \  {b \sb 
{4}}} -{{j \sp {2}} \  {a \sb {8}} \  {b \sb {3}}} -{{a \sb {4}} \  {b \sb 
{2}}}+{{a \sb {6}} \  {b \sb {1}}} 
\right)}
\  {| \sb {{ \  ik}}}}+{{\left( {{i \sp {2}} \  {a \sb {2}} \  {b \sb 
{8}}}+{{a \sb {1}} \  {b \sb {7}}} -{{i \sp {2}} \  {a \sb {5}} \  {b \sb 
{6}}}+{{i \sp {2}} \  {a \sb {6}} \  {b \sb {5}}}+{{a \sb {3}} \  {b \sb 
{4}}} -{{a \sb {4}} \  {b \sb {3}}}+{{i \sp {2}} \  {a \sb {8}} \  {b \sb 
{2}}}+{{a \sb {7}} \  {b \sb {1}}} 
\right)}
\  {| \sb {{ \  jk}}}}+{{\left( {{a \sb {1}} \  {b \sb {8}}}+{{a \sb {2}} \  
{b \sb {7}}} -{{a \sb {3}} \  {b \sb {6}}}+{{a \sb {4}} \  {b \sb {5}}}+{{a 
\sb {5}} \  {b \sb {4}}} -{{a \sb {6}} \  {b \sb {3}}}+{{a \sb {7}} \  {b \sb 
{2}}}+{{a \sb {8}} \  {b \sb {1}}} 
\right)}
\  {| \sb {{ \  ijk}}}} 
\leqno(31)
$$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
(32) -> test(
  ( I Y ) / 
  (  Y  ) = 
  ( Y I ) / _
  (  Y  ) )
$$
true 
\leqno(32)
$$
                                                                Type: Boolean
(33) -> U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)

$$
{{u \sp {{1, \: 1}}} \  {| \sp {{ \  1 \  1}}}}+{{u \sp {{1, \: 2}}} \  {| 
\sp {{ \  1 \  i}}}}+{{u \sp {{1, \: 3}}} \  {| \sp {{ \  1 \  j}}}}+{{u \sp 
{{1, \: 4}}} \  {| \sp {{ \  1 \  k}}}}+{{u \sp {{1, \: 5}}} \  {| \sp {{ \  
1 \  ij}}}}+{{u \sp {{1, \: 6}}} \  {| \sp {{ \  1 \  ik}}}}+{{u \sp {{1, \: 
7}}} \  {| \sp {{ \  1 \  jk}}}}+{{u \sp {{1, \: 8}}} \  {| \sp {{ \  1 \  
ijk}}}}+{{u \sp {{2, \: 1}}} \  {| \sp {{ \  i \  1}}}}+{{u \sp {{2, \: 2}}} 
\  {| \sp {{ \  i \  i}}}}+{{u \sp {{2, \: 3}}} \  {| \sp {{ \  i \  
j}}}}+{{u \sp {{2, \: 4}}} \  {| \sp {{ \  i \  k}}}}+{{u \sp {{2, \: 5}}} \  
{| \sp {{ \  i \  ij}}}}+{{u \sp {{2, \: 6}}} \  {| \sp {{ \  i \  ik}}}}+{{u 
\sp {{2, \: 7}}} \  {| \sp {{ \  i \  jk}}}}+{{u \sp {{2, \: 8}}} \  {| \sp 
{{ \  i \  ijk}}}}+{{u \sp {{3, \: 1}}} \  {| \sp {{ \  j \  1}}}}+{{u \sp 
{{3, \: 2}}} \  {| \sp {{ \  j \  i}}}}+{{u \sp {{3, \: 3}}} \  {| \sp {{ \  
j \  j}}}}+{{u \sp {{3, \: 4}}} \  {| \sp {{ \  j \  k}}}}+{{u \sp {{3, \: 
5}}} \  {| \sp {{ \  j \  ij}}}}+{{u \sp {{3, \: 6}}} \  {| \sp {{ \  j \  
ik}}}}+{{u \sp {{3, \: 7}}} \  {| \sp {{ \  j \  jk}}}}+{{u \sp {{3, \: 8}}} 
\  {| \sp {{ \  j \  ijk}}}}+{{u \sp {{4, \: 1}}} \  {| \sp {{ \  k \  
1}}}}+{{u \sp {{4, \: 2}}} \  {| \sp {{ \  k \  i}}}}+{{u \sp {{4, \: 3}}} \  
{| \sp {{ \  k \  j}}}}+{{u \sp {{4, \: 4}}} \  {| \sp {{ \  k \  k}}}}+{{u 
\sp {{4, \: 5}}} \  {| \sp {{ \  k \  ij}}}}+{{u \sp {{4, \: 6}}} \  {| \sp 
{{ \  k \  ik}}}}+{{u \sp {{4, \: 7}}} \  {| \sp {{ \  k \  jk}}}}+{{u \sp 
{{4, \: 8}}} \  {| \sp {{ \  k \  ijk}}}}+{{u \sp {{5, \: 1}}} \  {| \sp {{ \  
ij \  1}}}}+{{u \sp {{5, \: 2}}} \  {| \sp {{ \  ij \  i}}}}+{{u \sp {{5, \: 
3}}} \  {| \sp {{ \  ij \  j}}}}+{{u \sp {{5, \: 4}}} \  {| \sp {{ \  ij \  
k}}}}+{{u \sp {{5, \: 5}}} \  {| \sp {{ \  ij \  ij}}}}+{{u \sp {{5, \: 6}}} 
\  {| \sp {{ \  ij \  ik}}}}+{{u \sp {{5, \: 7}}} \  {| \sp {{ \  ij \  
jk}}}}+{{u \sp {{5, \: 8}}} \  {| \sp {{ \  ij \  ijk}}}}+{{u \sp {{6, \: 
1}}} \  {| \sp {{ \  ik \  1}}}}+{{u \sp {{6, \: 2}}} \  {| \sp {{ \  ik \  
i}}}}+{{u \sp {{6, \: 3}}} \  {| \sp {{ \  ik \  j}}}}+{{u \sp {{6, \: 4}}} \  
{| \sp {{ \  ik \  k}}}}+{{u \sp {{6, \: 5}}} \  {| \sp {{ \  ik \  
ij}}}}+{{u \sp {{6, \: 6}}} \  {| \sp {{ \  ik \  ik}}}}+{{u \sp {{6, \: 7}}} 
\  {| \sp {{ \  ik \  jk}}}}+{{u \sp {{6, \: 8}}} \  {| \sp {{ \  ik \  
ijk}}}}+{{u \sp {{7, \: 1}}} \  {| \sp {{ \  jk \  1}}}}+{{u \sp {{7, \: 2}}} 
\  {| \sp {{ \  jk \  i}}}}+{{u \sp {{7, \: 3}}} \  {| \sp {{ \  jk \  
j}}}}+{{u \sp {{7, \: 4}}} \  {| \sp {{ \  jk \  k}}}}+{{u \sp {{7, \: 5}}} \  
{| \sp {{ \  jk \  ij}}}}+{{u \sp {{7, \: 6}}} \  {| \sp {{ \  jk \  
ik}}}}+{{u \sp {{7, \: 7}}} \  {| \sp {{ \  jk \  jk}}}}+{{u \sp {{7, \: 8}}} 
\  {| \sp {{ \  jk \  ijk}}}}+{{u \sp {{8, \: 1}}} \  {| \sp {{ \  ijk \  
1}}}}+{{u \sp {{8, \: 2}}} \  {| \sp {{ \  ijk \  i}}}}+{{u \sp {{8, \: 3}}} 
\  {| \sp {{ \  ijk \  j}}}}+{{u \sp {{8, \: 4}}} \  {| \sp {{ \  ijk \  
k}}}}+{{u \sp {{8, \: 5}}} \  {| \sp {{ \  ijk \  ij}}}}+{{u \sp {{8, \: 6}}} 
\  {| \sp {{ \  ijk \  ik}}}}+{{u \sp {{8, \: 7}}} \  {| \sp {{ \  ijk \  
jk}}}}+{{u \sp {{8, \: 8}}} \  {| \sp {{ \  ijk \  ijk}}}} 
\leqno(33)
$$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
(34) -> ω:𝐋 :=                 
     (    Y I    )  /  
           U        -  
     (    I Y    )  /  
           U;
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
(35) -> Ú:=
   (  Y Λ  ) / 
   (   Y I ) / 
        V
$$
{8 \  {| \sp {{ \  1 \  1}}}}+{8 \  {i \sp {2}} \  {| \sp {{ \  i \  i}}}}+{8 
\  {j \sp {2}} \  {| \sp {{ \  j \  j}}}}+{8 \  {k \sp {2}} \  {| \sp {{ \  k 
\  k}}}} -{8 \  {i \sp {2}} \  {j \sp {2}} \  {| \sp {{ \  ij \  ij}}}} -{8 \  
{i \sp {2}} \  {k \sp {2}} \  {| \sp {{ \  ik \  ik}}}} -{8 \  {j \sp {2}} \  
{k \sp {2}} \  {| \sp {{ \  jk \  jk}}}} -{8 \  {i \sp {2}} \  {j \sp {2}} \  
{k \sp {2}} \  {| \sp {{ \  ijk \  ijk}}}} 
\leqno(35)
$$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
Ù:=
   (  Λ Y  ) / 
   ( I Y   ) / 
      V
$$
{8 \  {| \sp {{ \  1 \  1}}}}+{8 \  {i \sp {2}} \  {| \sp {{ \  i \  i}}}}+{8 
\  {j \sp {2}} \  {| \sp {{ \  j \  j}}}}+{8 \  {k \sp {2}} \  {| \sp {{ \  k 
\  k}}}} -{8 \  {i \sp {2}} \  {j \sp {2}} \  {| \sp {{ \  ij \  ij}}}} -{8 \  
{i \sp {2}} \  {k \sp {2}} \  {| \sp {{ \  ik \  ik}}}} -{8 \  {j \sp {2}} \  
{k \sp {2}} \  {| \sp {{ \  jk \  jk}}}} -{8 \  {i \sp {2}} \  {j \sp {2}} \  
{k \sp {2}} \  {| \sp {{ \  ijk \  ijk}}}} 
\leqno(36)
$$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
test(Ù=Ú)
$$
true 
\leqno(37)
$$
                                                                Type: Boolean
(38) -> Ũ := Ù

$$
{8 \  {| \sp {{ \  1 \  1}}}}+{8 \  {i \sp {2}} \  {| \sp {{ \  i \  i}}}}+{8 
\  {j \sp {2}} \  {| \sp {{ \  j \  j}}}}+{8 \  {k \sp {2}} \  {| \sp {{ \  k 
\  k}}}} -{8 \  {i \sp {2}} \  {j \sp {2}} \  {| \sp {{ \  ij \  ij}}}} -{8 \  
{i \sp {2}} \  {k \sp {2}} \  {| \sp {{ \  ik \  ik}}}} -{8 \  {j \sp {2}} \  
{k \sp {2}} \  {| \sp {{ \  jk \  jk}}}} -{8 \  {i \sp {2}} \  {j \sp {2}} \  
{k \sp {2}} \  {| \sp {{ \  ijk \  ijk}}}} 
\leqno(38)
$$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
test
     (    Y I    )  /
           Ũ        =
     (    I Y    )  /
           Ũ
$$
true 
\leqno(39)
$$
                                                                Type: Boolean
determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ), j,1..dim), i,1..dim)

$$
{16777216} \  {{i \sp {2}} \sp 4} \  {{j \sp {2}} \sp 4} \  {{k \sp {2}} \sp 
4} 
\leqno(40)
$$
                                                    Type: Expression(Integer)
(41) -> d:=ε1*𝐝.1+εi*𝐝.2+εj*𝐝.3+εk*𝐝.4+εij*𝐝.5+εik*𝐝.6+εjk*𝐝.7+εijk*𝐝.8

$$
{ε1 \  {| \sp {{ \  1}}}}+{εi \  {| \sp {{ \  i}}}}+{εj \  {| \sp {{ \  
j}}}}+{εk \  {| \sp {{ \  k}}}}+{εij \  {| \sp {{ \  ij}}}}+{εik \  {| \sp {{ 
\  ik}}}}+{εjk \  {| \sp {{ \  jk}}}}+{εijk \  {| \sp {{ \  ijk}}}} 
\leqno(41)
$$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
Ų:= Y/d
$$
{ε1 \  {| \sp {{ \  1 \  1}}}}+{εi \  {| \sp {{ \  1 \  i}}}}+{εj \  {| \sp 
{{ \  1 \  j}}}}+{εk \  {| \sp {{ \  1 \  k}}}}+{εij \  {| \sp {{ \  1 \  
ij}}}}+{εik \  {| \sp {{ \  1 \  ik}}}}+{εjk \  {| \sp {{ \  1 \  
jk}}}}+{εijk \  {| \sp {{ \  1 \  ijk}}}}+{εi \  {| \sp {{ \  i \  1}}}}+{{i 
\sp {2}} \  ε1 \  {| \sp {{ \  i \  i}}}}+{εij \  {| \sp {{ \  i \  
j}}}}+{εik \  {| \sp {{ \  i \  k}}}}+{{i \sp {2}} \  εj \  {| \sp {{ \  i \  
ij}}}}+{{i \sp {2}} \  εk \  {| \sp {{ \  i \  ik}}}}+{εijk \  {| \sp {{ \  i 
\  jk}}}}+{{i \sp {2}} \  εjk \  {| \sp {{ \  i \  ijk}}}}+{εj \  {| \sp {{ \  
j \  1}}}} -{εij \  {| \sp {{ \  j \  i}}}}+{{j \sp {2}} \  ε1 \  {| \sp {{ \  
j \  j}}}}+{εjk \  {| \sp {{ \  j \  k}}}} -{{j \sp {2}} \  εi \  {| \sp {{ \  
j \  ij}}}} -{εijk \  {| \sp {{ \  j \  ik}}}}+{{j \sp {2}} \  εk \  {| \sp 
{{ \  j \  jk}}}} -{{j \sp {2}} \  εik \  {| \sp {{ \  j \  ijk}}}}+{εk \  {| 
\sp {{ \  k \  1}}}} -{εik \  {| \sp {{ \  k \  i}}}} -{εjk \  {| \sp {{ \  k 
\  j}}}}+{{k \sp {2}} \  ε1 \  {| \sp {{ \  k \  k}}}}+{εijk \  {| \sp {{ \  
k \  ij}}}} -{{k \sp {2}} \  εi \  {| \sp {{ \  k \  ik}}}} -{{k \sp {2}} \  
εj \  {| \sp {{ \  k \  jk}}}}+{{k \sp {2}} \  εij \  {| \sp {{ \  k \  
ijk}}}}+{εij \  {| \sp {{ \  ij \  1}}}} -{{i \sp {2}} \  εj \  {| \sp {{ \  
ij \  i}}}}+{{j \sp {2}} \  εi \  {| \sp {{ \  ij \  j}}}}+{εijk \  {| \sp {{ 
\  ij \  k}}}} -{{i \sp {2}} \  {j \sp {2}} \  ε1 \  {| \sp {{ \  ij \  
ij}}}} -{{i \sp {2}} \  εjk \  {| \sp {{ \  ij \  ik}}}}+{{j \sp {2}} \  εik 
\  {| \sp {{ \  ij \  jk}}}} -{{i \sp {2}} \  {j \sp {2}} \  εk \  {| \sp {{ 
\  ij \  ijk}}}}+{εik \  {| \sp {{ \  ik \  1}}}} -{{i \sp {2}} \  εk \  {| 
\sp {{ \  ik \  i}}}} -{εijk \  {| \sp {{ \  ik \  j}}}}+{{k \sp {2}} \  εi \  
{| \sp {{ \  ik \  k}}}}+{{i \sp {2}} \  εjk \  {| \sp {{ \  ik \  ij}}}} 
-{{i \sp {2}} \  {k \sp {2}} \  ε1 \  {| \sp {{ \  ik \  ik}}}} -{{k \sp {2}} 
\  εij \  {| \sp {{ \  ik \  jk}}}}+{{i \sp {2}} \  {k \sp {2}} \  εj \  {| 
\sp {{ \  ik \  ijk}}}}+{εjk \  {| \sp {{ \  jk \  1}}}}+{εijk \  {| \sp {{ \  
jk \  i}}}} -{{j \sp {2}} \  εk \  {| \sp {{ \  jk \  j}}}}+{{k \sp {2}} \  
εj \  {| \sp {{ \  jk \  k}}}} -{{j \sp {2}} \  εik \  {| \sp {{ \  jk \  
ij}}}}+{{k \sp {2}} \  εij \  {| \sp {{ \  jk \  ik}}}} -{{j \sp {2}} \  {k 
\sp {2}} \  ε1 \  {| \sp {{ \  jk \  jk}}}} -{{j \sp {2}} \  {k \sp {2}} \  
εi \  {| \sp {{ \  jk \  ijk}}}}+{εijk \  {| \sp {{ \  ijk \  1}}}}+{{i \sp 
{2}} \  εjk \  {| \sp {{ \  ijk \  i}}}} -{{j \sp {2}} \  εik \  {| \sp {{ \  
ijk \  j}}}}+{{k \sp {2}} \  εij \  {| \sp {{ \  ijk \  k}}}} -{{i \sp {2}} \  
{j \sp {2}} \  εk \  {| \sp {{ \  ijk \  ij}}}}+{{i \sp {2}} \  {k \sp {2}} \  
εj \  {| \sp {{ \  ijk \  ik}}}} -{{j \sp {2}} \  {k \sp {2}} \  εi \  {| \sp 
{{ \  ijk \  jk}}}} -{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  {| 
\sp {{ \  ijk \  ijk}}}} 
\leqno(42)
$$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
(43) -> u1:=matrix Ξ(Ξ(retract((𝐞.i 𝐞.j)/Ų), i,1..dim), j,1..dim)
$$
\left[
\begin{array}{cccccccc}
ε1 & εi & εj & εk & εij & εik & εjk & εijk \ 
εi & {{i \sp {2}} \  ε1} & -εij & -εik & -{{i \sp {2}} \  εj} & -{{i \sp {2}} 
\  εk} & εijk & {{i \sp {2}} \  εjk} \ 
εj & εij & {{j \sp {2}} \  ε1} & -εjk & {{j \sp {2}} \  εi} & -εijk & -{{j 
\sp {2}} \  εk} & -{{j \sp {2}} \  εik} \ 
εk & εik & εjk & {{k \sp {2}} \  ε1} & εijk & {{k \sp {2}} \  εi} & {{k \sp 
{2}} \  εj} & {{k \sp {2}} \  εij} \ 
εij & {{i \sp {2}} \  εj} & -{{j \sp {2}} \  εi} & εijk & -{{i \sp {2}} \  {j 
\sp {2}} \  ε1} & {{i \sp {2}} \  εjk} & -{{j \sp {2}} \  εik} & -{{i \sp 
{2}} \  {j \sp {2}} \  εk} \ 
εik & {{i \sp {2}} \  εk} & -εijk & -{{k \sp {2}} \  εi} & -{{i \sp {2}} \  
εjk} & -{{i \sp {2}} \  {k \sp {2}} \  ε1} & {{k \sp {2}} \  εij} & {{i \sp 
{2}} \  {k \sp {2}} \  εj} \ 
εjk & εijk & {{j \sp {2}} \  εk} & -{{k \sp {2}} \  εj} & {{j \sp {2}} \  
εik} & -{{k \sp {2}} \  εij} & -{{j \sp {2}} \  {k \sp {2}} \  ε1} & -{{j \sp 
{2}} \  {k \sp {2}} \  εi} \ 
εijk & {{i \sp {2}} \  εjk} & -{{j \sp {2}} \  εik} & {{k \sp {2}} \  εij} & 
-{{i \sp {2}} \  {j \sp {2}} \  εk} & {{i \sp {2}} \  {k \sp {2}} \  εj} & 
-{{j \sp {2}} \  {k \sp {2}} \  εi} & -{{i \sp {2}} \  {j \sp {2}} \  {k \sp 
{2}} \  ε1} 
\end{array}
\right]
\leqno(43)
$$
                                            Type: Matrix(Expression(Integer))
(44) -> --Ů:=determinant u1
--factor(numer Ů)/factor(denom Ů)
1

$$
1 
\leqno(44)
$$
                                                        Type: PositiveInteger
(45) -> ck:=solve(equate(Ũ=Ų),[ε1,εi,εj,εk,εij,εik,εjk,εijk]).1
   Compiling function equate with type Equation(ClosedLinearOperator(
      OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)))
       -> List(Equation(Expression(Integer))) 

$$
\left[
{ε1=8}, \: {εi=0}, \: {εj=0}, \: {εk=0}, \: {εij=0}, \: {εik=0}, \: {εjk=0}, 
\: {εijk=0} 
\right]
\leqno(45)
$$
                                    Type: List(Equation(Expression(Integer)))
(46) -> a:=sb('a,[1])*i+sb('a,[2])*j+sb('a,[3])*k

$$
{{a \sb {1}} \  {| \sb {{ \  i}}}}+{{a \sb {2}} \  {| \sb {{ \  j}}}}+{{a \sb 
{3}} \  {| \sb {{ \  k}}}} 
\leqno(46)
$$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
b:=sb('b,[1])*i+sb('b,[2])*j+sb('b,[3])*k
$$
{{b \sb {1}} \  {| \sb {{ \  i}}}}+{{b \sb {2}} \  {| \sb {{ \  j}}}}+{{b \sb 
{3}} \  {| \sb {{ \  k}}}} 
\leqno(47)
$$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
(a,a)/Ų
$$
{\left( {{k \sp {2}} \  {{a \sb {3}} \sp 2}}+{{j \sp {2}} \  {{a \sb {2}} \sp 
2}}+{{i \sp {2}} \  {{a \sb {1}} \sp 2}} 
\right)}
\  ε1 
\leqno(48)
$$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
(b,b)/Ų
$$
{\left( {{k \sp {2}} \  {{b \sb {3}} \sp 2}}+{{j \sp {2}} \  {{b \sb {2}} \sp 
2}}+{{i \sp {2}} \  {{b \sb {1}} \sp 2}} 
\right)}
\  ε1 
\leqno(49)
$$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
(a,b)/Ų
$$
{{\left( {{a \sb {2}} \  {b \sb {3}}} -{{a \sb {3}} \  {b \sb {2}}} 
\right)}
\  εjk}+{{\left( {{a \sb {1}} \  {b \sb {3}}} -{{a \sb {3}} \  {b \sb {1}}} 
\right)}
\  εik}+{{\left( {{a \sb {1}} \  {b \sb {2}}} -{{a \sb {2}} \  {b \sb {1}}} 
\right)}
\  εij}+{{\left( {{k \sp {2}} \  {a \sb {3}} \  {b \sb {3}}}+{{j \sp {2}} \  
{a \sb {2}} \  {b \sb {2}}}+{{i \sp {2}} \  {a \sb {1}} \  {b \sb {1}}} 
\right)}
\  ε1} 
\leqno(50)
$$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
(51) -> mU:=inverse matrix Ξ(Ξ(retract((𝐞.i*𝐞.j)/Ų), i,1..dim), j,1..dim);
                                 Type: Union(Matrix(Expression(Integer)),...)
Ω:=Σ(Σ(mU(i,j)*(𝐞.i*𝐞.j), i,1..dim), j,1..dim);

Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
ΩX:=Ω/X;
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
(54) -> --squareFreePart factor denom Ů / squareFreePart factor numer Ů
matrix Ξ(Ξ(numer retract(Ω/(𝐝.i*𝐝.j)), i,1..dim), j,1..dim)
$$
\left[
\begin{array}{cccccccc}
{-{{{i \sp {2}} \sp 2} \  {{j \sp {2}} \sp 2} \  {k \sp {2}} \  ε1 \  {εk \sp 
2}} -{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εij \  εijk \  
εk}+{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  {εjk \sp 2}} 
-{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εi \  εijk \  εjk} -{{{i 
\sp {2}} \sp 2} \  {j \sp {2}} \  {{k \sp {2}} \sp 2} \  ε1 \  {εj \sp 2}}+{2 
\  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εijk \  εik \  εj}+{{i \sp 
{2}} \  {{j \sp {2}} \sp 2} \  {k \sp {2}} \  ε1 \  {εik \sp 2}}+{{i \sp {2}} 
\  {j \sp {2}} \  {k \sp {2}} \  ε1 \  {εijk \sp 2}}+{{i \sp {2}} \  {j \sp 
{2}} \  {{k \sp {2}} \sp 2} \  ε1 \  {εij \sp 2}} -{{i \sp {2}} \  {{j \sp 
{2}} \sp 2} \  {{k \sp {2}} \sp 2} \  ε1 \  {εi \sp 2}}+{{{i \sp {2}} \sp 2} 
\  {{j \sp {2}} \sp 2} \  {{k \sp {2}} \sp 2} \  {ε1 \sp 3}}} & {{{i \sp {2}} 
\  {{j \sp {2}} \sp 2} \  {k \sp {2}} \  εi \  {εk \sp 2}}+{2 \  {i \sp {2}} 
\  {j \sp {2}} \  {k \sp {2}} \  εij \  εjk \  εk}+{{i \sp {2}} \  {j \sp 
{2}} \  {k \sp {2}} \  εi \  {εjk \sp 2}}+{{\left( -{2 \  {i \sp {2}} \  {j 
\sp {2}} \  {k \sp {2}} \  εik \  εj} -{2 \  {i \sp {2}} \  {j \sp {2}} \  {k 
\sp {2}} \  ε1 \  εijk} 
\right)}
\  εjk}+{{i \sp {2}} \  {j \sp {2}} \  {{k \sp {2}} \sp 2} \  εi \  {εj \sp 
2}} -{{{j \sp {2}} \sp 2} \  {k \sp {2}} \  εi \  {εik \sp 2}}+{{j \sp {2}} \  
{k \sp {2}} \  εi \  {εijk \sp 2}} -{{j \sp {2}} \  {{k \sp {2}} \sp 2} \  εi 
\  {εij \sp 2}}+{{{j \sp {2}} \sp 2} \  {{k \sp {2}} \sp 2} \  {εi \sp 3}} 
-{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  {{k \sp {2}} \sp 2} \  {ε1 \sp 2} \  
εi}} & {{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  {k \sp {2}} \  εj \  {εk \sp 
2}} -{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εij \  εik \  εk} 
-{{{i \sp {2}} \sp 2} \  {k \sp {2}} \  εj \  {εjk \sp 2}} -{2 \  {i \sp {2}} 
\  {j \sp {2}} \  {k \sp {2}} \  εi \  εik \  εjk}+{{{i \sp {2}} \sp 2} \  
{{k \sp {2}} \sp 2} \  {εj \sp 3}}+{{\left( {{i \sp {2}} \  {j \sp {2}} \  {k 
\sp {2}} \  {εik \sp 2}}+{{i \sp {2}} \  {k \sp {2}} \  {εijk \sp 2}} -{{i 
\sp {2}} \  {{k \sp {2}} \sp 2} \  {εij \sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  
{{k \sp {2}} \sp 2} \  {εi \sp 2}} -{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  
{{k \sp {2}} \sp 2} \  {ε1 \sp 2}} 
\right)}
\  εj}+{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εijk \  εik}} 
& {{{{i \sp {2}} \sp 2} \  {{j \sp {2}} \sp 2} \  {εk \sp 3}}+{{\left( -{{{i 
\sp {2}} \sp 2} \  {j \sp {2}} \  {εjk \sp 2}}+{{{i \sp {2}} \sp 2} \  {j \sp 
{2}} \  {k \sp {2}} \  {εj \sp 2}} -{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  
{εik \sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {εijk \sp 2}}+{{i \sp {2}} \  {j 
\sp {2}} \  {k \sp {2}} \  {εij \sp 2}}+{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  
{k \sp {2}} \  {εi \sp 2}} -{{{i \sp {2}} \sp 2} \  {{j \sp {2}} \sp 2} \  {k 
\sp {2}} \  {ε1 \sp 2}} 
\right)}
\  εk}+{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εi \  εij \  εjk} 
-{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εij \  εik \  εj} -{2 \  
{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εij \  εijk}} & {{{i \sp 
{2}} \  {j \sp {2}} \  {k \sp {2}} \  εij \  {εk \sp 2}}+{{\left( {2 \  {i 
\sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εi \  εjk} -{2 \  {i \sp {2}} \  {j 
\sp {2}} \  {k \sp {2}} \  εik \  εj} -{2 \  {i \sp {2}} \  {j \sp {2}} \  {k 
\sp {2}} \  ε1 \  εijk} 
\right)}
\  εk}+{{i \sp {2}} \  {k \sp {2}} \  εij \  {εjk \sp 2}} -{{i \sp {2}} \  
{{k \sp {2}} \sp 2} \  εij \  {εj \sp 2}}+{{j \sp {2}} \  {k \sp {2}} \  εij 
\  {εik \sp 2}} -{{k \sp {2}} \  εij \  {εijk \sp 2}}+{{{k \sp {2}} \sp 2} \  
{εij \sp 3}}+{{\left( -{{j \sp {2}} \  {{k \sp {2}} \sp 2} \  {εi \sp 2}}+{{i 
\sp {2}} \  {j \sp {2}} \  {{k \sp {2}} \sp 2} \  {ε1 \sp 2}} 
\right)}
\  εij}} & {-{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  εik \  {εk \sp 2}} -{2 \  
{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εij \  εj \  εk}+{{i \sp {2}} \  
{j \sp {2}} \  εik \  {εjk \sp 2}} -{2 \  {i \sp {2}} \  {j \sp {2}} \  {k 
\sp {2}} \  εi \  εj \  εjk}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  
εik \  {εj \sp 2}}+{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  
εijk \  εj}+{{{j \sp {2}} \sp 2} \  {εik \sp 3}}+{{\left( -{{j \sp {2}} \  
{εijk \sp 2}}+{{j \sp {2}} \  {k \sp {2}} \  {εij \sp 2}} -{{{j \sp {2}} \sp 
2} \  {k \sp {2}} \  {εi \sp 2}}+{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  {k 
\sp {2}} \  {ε1 \sp 2}} 
\right)}
\  εik}} & {-{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  εjk \  {εk \sp 2}}+{2 \  
{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εi \  εij \  εk}+{{{i \sp {2}} 
\sp 2} \  {εjk \sp 3}}+{{\left( -{{{i \sp {2}} \sp 2} \  {k \sp {2}} \  {εj 
\sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {εik \sp 2}} -{{i \sp {2}} \  {εijk 
\sp 2}}+{{i \sp {2}} \  {k \sp {2}} \  {εij \sp 2}}+{{i \sp {2}} \  {j \sp 
{2}} \  {k \sp {2}} \  {εi \sp 2}}+{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  {k 
\sp {2}} \  {ε1 \sp 2}} 
\right)}
\  εjk} -{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εi \  εik \  εj} 
-{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εi \  εijk}} & {{{i 
\sp {2}} \  {j \sp {2}} \  εijk \  {εk \sp 2}} -{2 \  {i \sp {2}} \  {j \sp 
{2}} \  {k \sp {2}} \  ε1 \  εij \  εk} -{{i \sp {2}} \  εijk \  {εjk \sp 2}} 
-{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εi \  εjk}+{{i \sp 
{2}} \  {k \sp {2}} \  εijk \  {εj \sp 2}}+{2 \  {i \sp {2}} \  {j \sp {2}} \  
{k \sp {2}} \  ε1 \  εik \  εj} -{{j \sp {2}} \  εijk \  {εik \sp 2}}+{εijk 
\sp 3}+{{\left( -{{k \sp {2}} \  {εij \sp 2}}+{{j \sp {2}} \  {k \sp {2}} \  
{εi \sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {ε1 \sp 2}} 
\right)}
\  εijk}} \ 
{{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  {k \sp {2}} \  εi \  {εk \sp 2}}+{2 \  
{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εij \  εjk \  εk}+{{i \sp {2}} \  
{j \sp {2}} \  {k \sp {2}} \  εi \  {εjk \sp 2}}+{{\left( -{2 \  {i \sp {2}} 
\  {j \sp {2}} \  {k \sp {2}} \  εik \  εj} -{2 \  {i \sp {2}} \  {j \sp {2}} 
\  {k \sp {2}} \  ε1 \  εijk} 
\right)}
\  εjk}+{{i \sp {2}} \  {j \sp {2}} \  {{k \sp {2}} \sp 2} \  εi \  {εj \sp 
2}} -{{{j \sp {2}} \sp 2} \  {k \sp {2}} \  εi \  {εik \sp 2}}+{{j \sp {2}} \  
{k \sp {2}} \  εi \  {εijk \sp 2}} -{{j \sp {2}} \  {{k \sp {2}} \sp 2} \  εi 
\  {εij \sp 2}}+{{{j \sp {2}} \sp 2} \  {{k \sp {2}} \sp 2} \  {εi \sp 3}} 
-{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  {{k \sp {2}} \sp 2} \  {ε1 \sp 2} \  
εi}} & {-{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  {k \sp {2}} \  ε1 \  {εk \sp 
2}} -{2 \  {j \sp {2}} \  {k \sp {2}} \  εij \  εijk \  εk}+{{i \sp {2}} \  
{j \sp {2}} \  {k \sp {2}} \  ε1 \  {εjk \sp 2}} -{2 \  {j \sp {2}} \  {k \sp 
{2}} \  εi \  εijk \  εjk} -{{i \sp {2}} \  {j \sp {2}} \  {{k \sp {2}} \sp 
2} \  ε1 \  {εj \sp 2}}+{2 \  {j \sp {2}} \  {k \sp {2}} \  εijk \  εik \  
εj}+{{{j \sp {2}} \sp 2} \  {k \sp {2}} \  ε1 \  {εik \sp 2}}+{{j \sp {2}} \  
{k \sp {2}} \  ε1 \  {εijk \sp 2}}+{{j \sp {2}} \  {{k \sp {2}} \sp 2} \  ε1 
\  {εij \sp 2}} -{{{j \sp {2}} \sp 2} \  {{k \sp {2}} \sp 2} \  ε1 \  {εi \sp 
2}}+{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  {{k \sp {2}} \sp 2} \  {ε1 \sp 
3}}} & {-{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εij \  {εk \sp 
2}}+{{\left( -{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εi \  
εjk}+{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εik \  εj}+{2 \  {i 
\sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εijk} 
\right)}
\  εk} -{{i \sp {2}} \  {k \sp {2}} \  εij \  {εjk \sp 2}}+{{i \sp {2}} \  
{{k \sp {2}} \sp 2} \  εij \  {εj \sp 2}} -{{j \sp {2}} \  {k \sp {2}} \  εij 
\  {εik \sp 2}}+{{k \sp {2}} \  εij \  {εijk \sp 2}} -{{{k \sp {2}} \sp 2} \  
{εij \sp 3}}+{{\left( {{j \sp {2}} \  {{k \sp {2}} \sp 2} \  {εi \sp 2}} -{{i 
\sp {2}} \  {j \sp {2}} \  {{k \sp {2}} \sp 2} \  {ε1 \sp 2}} 
\right)}
\  εij}} & {{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  εik \  {εk \sp 2}}+{2 \  
{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εij \  εj \  εk} -{{i \sp {2}} \  
{j \sp {2}} \  εik \  {εjk \sp 2}}+{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp 
{2}} \  εi \  εj \  εjk} -{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εik \  
{εj \sp 2}} -{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εijk \  
εj} -{{{j \sp {2}} \sp 2} \  {εik \sp 3}}+{{\left( {{j \sp {2}} \  {εijk \sp 
2}} -{{j \sp {2}} \  {k \sp {2}} \  {εij \sp 2}}+{{{j \sp {2}} \sp 2} \  {k 
\sp {2}} \  {εi \sp 2}} -{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  {k \sp {2}} \  
{ε1 \sp 2}} 
\right)}
\  εik}} & {-{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εj \  {εk \sp 
2}}+{2 \  {j \sp {2}} \  {k \sp {2}} \  εij \  εik \  εk}+{{i \sp {2}} \  {k 
\sp {2}} \  εj \  {εjk \sp 2}}+{2 \  {j \sp {2}} \  {k \sp {2}} \  εi \  εik 
\  εjk} -{{i \sp {2}} \  {{k \sp {2}} \sp 2} \  {εj \sp 3}}+{{\left( -{{j \sp 
{2}} \  {k \sp {2}} \  {εik \sp 2}} -{{k \sp {2}} \  {εijk \sp 2}}+{{{k \sp 
{2}} \sp 2} \  {εij \sp 2}} -{{j \sp {2}} \  {{k \sp {2}} \sp 2} \  {εi \sp 
2}}+{{i \sp {2}} \  {j \sp {2}} \  {{k \sp {2}} \sp 2} \  {ε1 \sp 2}} 
\right)}
\  εj} -{2 \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εijk \  εik}} & {-{{i \sp 
{2}} \  {{j \sp {2}} \sp 2} \  {εk \sp 3}}+{{\left( {{i \sp {2}} \  {j \sp 
{2}} \  {εjk \sp 2}} -{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {εj \sp 
2}}+{{{j \sp {2}} \sp 2} \  {εik \sp 2}} -{{j \sp {2}} \  {εijk \sp 2}} -{{j 
\sp {2}} \  {k \sp {2}} \  {εij \sp 2}} -{{{j \sp {2}} \sp 2} \  {k \sp {2}} 
\  {εi \sp 2}}+{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  {k \sp {2}} \  {ε1 \sp 
2}} 
\right)}
\  εk} -{2 \  {j \sp {2}} \  {k \sp {2}} \  εi \  εij \  εjk}+{2 \  {j \sp 
{2}} \  {k \sp {2}} \  εij \  εik \  εj}+{2 \  {j \sp {2}} \  {k \sp {2}} \  
ε1 \  εij \  εijk}} & {{{i \sp {2}} \  {j \sp {2}} \  εijk \  {εk \sp 2}} -{2 
\  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εij \  εk} -{{i \sp 
{2}} \  εijk \  {εjk \sp 2}} -{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} 
\  ε1 \  εi \  εjk}+{{i \sp {2}} \  {k \sp {2}} \  εijk \  {εj \sp 2}}+{2 \  
{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εik \  εj} -{{j \sp {2}} \  
εijk \  {εik \sp 2}}+{εijk \sp 3}+{{\left( -{{k \sp {2}} \  {εij \sp 2}}+{{j 
\sp {2}} \  {k \sp {2}} \  {εi \sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp 
{2}} \  {ε1 \sp 2}} 
\right)}
\  εijk}} & {-{{i \sp {2}} \  {j \sp {2}} \  εjk \  {εk \sp 2}}+{2 \  {j \sp 
{2}} \  {k \sp {2}} \  εi \  εij \  εk}+{{i \sp {2}} \  {εjk \sp 3}}+{{\left( 
-{{i \sp {2}} \  {k \sp {2}} \  {εj \sp 2}}+{{j \sp {2}} \  {εik \sp 2}} 
-{εijk \sp 2}+{{k \sp {2}} \  {εij \sp 2}}+{{j \sp {2}} \  {k \sp {2}} \  {εi 
\sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {ε1 \sp 2}} 
\right)}
\  εjk} -{2 \  {j \sp {2}} \  {k \sp {2}} \  εi \  εik \  εj} -{2 \  {j \sp 
{2}} \  {k \sp {2}} \  ε1 \  εi \  εijk}} \ 
{{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  {k \sp {2}} \  εj \  {εk \sp 2}} -{2 
\  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εij \  εik \  εk} -{{{i \sp 
{2}} \sp 2} \  {k \sp {2}} \  εj \  {εjk \sp 2}} -{2 \  {i \sp {2}} \  {j \sp 
{2}} \  {k \sp {2}} \  εi \  εik \  εjk}+{{{i \sp {2}} \sp 2} \  {{k \sp {2}} 
\sp 2} \  {εj \sp 3}}+{{\left( {{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  
{εik \sp 2}}+{{i \sp {2}} \  {k \sp {2}} \  {εijk \sp 2}} -{{i \sp {2}} \  
{{k \sp {2}} \sp 2} \  {εij \sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {{k \sp 
{2}} \sp 2} \  {εi \sp 2}} -{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  {{k \sp 
{2}} \sp 2} \  {ε1 \sp 2}} 
\right)}
\  εj}+{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εijk \  εik}} 
& {{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εij \  {εk \sp 2}}+{{\left( 
{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εi \  εjk} -{2 \  {i \sp 
{2}} \  {j \sp {2}} \  {k \sp {2}} \  εik \  εj} -{2 \  {i \sp {2}} \  {j \sp 
{2}} \  {k \sp {2}} \  ε1 \  εijk} 
\right)}
\  εk}+{{i \sp {2}} \  {k \sp {2}} \  εij \  {εjk \sp 2}} -{{i \sp {2}} \  
{{k \sp {2}} \sp 2} \  εij \  {εj \sp 2}}+{{j \sp {2}} \  {k \sp {2}} \  εij 
\  {εik \sp 2}} -{{k \sp {2}} \  εij \  {εijk \sp 2}}+{{{k \sp {2}} \sp 2} \  
{εij \sp 3}}+{{\left( -{{j \sp {2}} \  {{k \sp {2}} \sp 2} \  {εi \sp 2}}+{{i 
\sp {2}} \  {j \sp {2}} \  {{k \sp {2}} \sp 2} \  {ε1 \sp 2}} 
\right)}
\  εij}} & {-{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  {εk 
\sp 2}} -{2 \  {i \sp {2}} \  {k \sp {2}} \  εij \  εijk \  εk}+{{{i \sp {2}} 
\sp 2} \  {k \sp {2}} \  ε1 \  {εjk \sp 2}} -{2 \  {i \sp {2}} \  {k \sp {2}} 
\  εi \  εijk \  εjk} -{{{i \sp {2}} \sp 2} \  {{k \sp {2}} \sp 2} \  ε1 \  
{εj \sp 2}}+{2 \  {i \sp {2}} \  {k \sp {2}} \  εijk \  εik \  εj}+{{i \sp 
{2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  {εik \sp 2}}+{{i \sp {2}} \  {k 
\sp {2}} \  ε1 \  {εijk \sp 2}}+{{i \sp {2}} \  {{k \sp {2}} \sp 2} \  ε1 \  
{εij \sp 2}} -{{i \sp {2}} \  {j \sp {2}} \  {{k \sp {2}} \sp 2} \  ε1 \  {εi 
\sp 2}}+{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  {{k \sp {2}} \sp 2} \  {ε1 \sp 
3}}} & {{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  εjk \  {εk \sp 2}} -{2 \  {i 
\sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εi \  εij \  εk} -{{{i \sp {2}} \sp 
2} \  {εjk \sp 3}}+{{\left( {{{i \sp {2}} \sp 2} \  {k \sp {2}} \  {εj \sp 
2}} -{{i \sp {2}} \  {j \sp {2}} \  {εik \sp 2}}+{{i \sp {2}} \  {εijk \sp 
2}} -{{i \sp {2}} \  {k \sp {2}} \  {εij \sp 2}} -{{i \sp {2}} \  {j \sp {2}} 
\  {k \sp {2}} \  {εi \sp 2}} -{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  {k \sp 
{2}} \  {ε1 \sp 2}} 
\right)}
\  εjk}+{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εi \  εik \  
εj}+{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εi \  εijk}} & 
{{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εi \  {εk \sp 2}}+{2 \  {i \sp 
{2}} \  {k \sp {2}} \  εij \  εjk \  εk}+{{i \sp {2}} \  {k \sp {2}} \  εi \  
{εjk \sp 2}}+{{\left( -{2 \  {i \sp {2}} \  {k \sp {2}} \  εik \  εj} -{2 \  
{i \sp {2}} \  {k \sp {2}} \  ε1 \  εijk} 
\right)}
\  εjk}+{{i \sp {2}} \  {{k \sp {2}} \sp 2} \  εi \  {εj \sp 2}} -{{j \sp 
{2}} \  {k \sp {2}} \  εi \  {εik \sp 2}}+{{k \sp {2}} \  εi \  {εijk \sp 2}} 
-{{{k \sp {2}} \sp 2} \  εi \  {εij \sp 2}}+{{j \sp {2}} \  {{k \sp {2}} \sp 
2} \  {εi \sp 3}} -{{i \sp {2}} \  {j \sp {2}} \  {{k \sp {2}} \sp 2} \  {ε1 
\sp 2} \  εi}} & {-{{i \sp {2}} \  {j \sp {2}} \  εijk \  {εk \sp 2}}+{2 \  
{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εij \  εk}+{{i \sp {2}} \  
εijk \  {εjk \sp 2}}+{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  
εi \  εjk} -{{i \sp {2}} \  {k \sp {2}} \  εijk \  {εj \sp 2}} -{2 \  {i \sp 
{2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εik \  εj}+{{j \sp {2}} \  εijk \  
{εik \sp 2}} -{εijk \sp 3}+{{\left( {{k \sp {2}} \  {εij \sp 2}} -{{j \sp 
{2}} \  {k \sp {2}} \  {εi \sp 2}} -{{i \sp {2}} \  {j \sp {2}} \  {k \sp 
{2}} \  {ε1 \sp 2}} 
\right)}
\  εijk}} & {-{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  {εk \sp 3}}+{{\left( 
{{{i \sp {2}} \sp 2} \  {εjk \sp 2}} -{{{i \sp {2}} \sp 2} \  {k \sp {2}} \  
{εj \sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {εik \sp 2}} -{{i \sp {2}} \  
{εijk \sp 2}} -{{i \sp {2}} \  {k \sp {2}} \  {εij \sp 2}} -{{i \sp {2}} \  
{j \sp {2}} \  {k \sp {2}} \  {εi \sp 2}}+{{{i \sp {2}} \sp 2} \  {j \sp {2}} 
\  {k \sp {2}} \  {ε1 \sp 2}} 
\right)}
\  εk} -{2 \  {i \sp {2}} \  {k \sp {2}} \  εi \  εij \  εjk}+{2 \  {i \sp 
{2}} \  {k \sp {2}} \  εij \  εik \  εj}+{2 \  {i \sp {2}} \  {k \sp {2}} \  
ε1 \  εij \  εijk}} & {{{i \sp {2}} \  {j \sp {2}} \  εik \  {εk \sp 2}}+{2 \  
{i \sp {2}} \  {k \sp {2}} \  εij \  εj \  εk} -{{i \sp {2}} \  εik \  {εjk 
\sp 2}}+{2 \  {i \sp {2}} \  {k \sp {2}} \  εi \  εj \  εjk} -{{i \sp {2}} \  
{k \sp {2}} \  εik \  {εj \sp 2}} -{2 \  {i \sp {2}} \  {k \sp {2}} \  ε1 \  
εijk \  εj} -{{j \sp {2}} \  {εik \sp 3}}+{{\left( {εijk \sp 2} -{{k \sp {2}} 
\  {εij \sp 2}}+{{j \sp {2}} \  {k \sp {2}} \  {εi \sp 2}} -{{i \sp {2}} \  
{j \sp {2}} \  {k \sp {2}} \  {ε1 \sp 2}} 
\right)}
\  εik}} \ 
{{{{i \sp {2}} \sp 2} \  {{j \sp {2}} \sp 2} \  {εk \sp 3}}+{{\left( -{{{i 
\sp {2}} \sp 2} \  {j \sp {2}} \  {εjk \sp 2}}+{{{i \sp {2}} \sp 2} \  {j \sp 
{2}} \  {k \sp {2}} \  {εj \sp 2}} -{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  
{εik \sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {εijk \sp 2}}+{{i \sp {2}} \  {j 
\sp {2}} \  {k \sp {2}} \  {εij \sp 2}}+{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  
{k \sp {2}} \  {εi \sp 2}} -{{{i \sp {2}} \sp 2} \  {{j \sp {2}} \sp 2} \  {k 
\sp {2}} \  {ε1 \sp 2}} 
\right)}
\  εk}+{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εi \  εij \  εjk} 
-{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εij \  εik \  εj} -{2 \  
{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εij \  εijk}} & {-{{i \sp 
{2}} \  {{j \sp {2}} \sp 2} \  εik \  {εk \sp 2}} -{2 \  {i \sp {2}} \  {j 
\sp {2}} \  {k \sp {2}} \  εij \  εj \  εk}+{{i \sp {2}} \  {j \sp {2}} \  
εik \  {εjk \sp 2}} -{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εi \  
εj \  εjk}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εik \  {εj \sp 
2}}+{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εijk \  εj}+{{{j 
\sp {2}} \sp 2} \  {εik \sp 3}}+{{\left( -{{j \sp {2}} \  {εijk \sp 2}}+{{j 
\sp {2}} \  {k \sp {2}} \  {εij \sp 2}} -{{{j \sp {2}} \sp 2} \  {k \sp {2}} 
\  {εi \sp 2}}+{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  {k \sp {2}} \  {ε1 \sp 
2}} 
\right)}
\  εik}} & {-{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  εjk \  {εk \sp 2}}+{2 \  
{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εi \  εij \  εk}+{{{i \sp {2}} 
\sp 2} \  {εjk \sp 3}}+{{\left( -{{{i \sp {2}} \sp 2} \  {k \sp {2}} \  {εj 
\sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {εik \sp 2}} -{{i \sp {2}} \  {εijk 
\sp 2}}+{{i \sp {2}} \  {k \sp {2}} \  {εij \sp 2}}+{{i \sp {2}} \  {j \sp 
{2}} \  {k \sp {2}} \  {εi \sp 2}}+{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  {k 
\sp {2}} \  {ε1 \sp 2}} 
\right)}
\  εjk} -{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εi \  εik \  εj} 
-{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εi \  εijk}} & 
{-{{{i \sp {2}} \sp 2} \  {{j \sp {2}} \sp 2} \  ε1 \  {εk \sp 2}} -{2 \  {i 
\sp {2}} \  {j \sp {2}} \  εij \  εijk \  εk}+{{{i \sp {2}} \sp 2} \  {j \sp 
{2}} \  ε1 \  {εjk \sp 2}} -{2 \  {i \sp {2}} \  {j \sp {2}} \  εi \  εijk \  
εjk} -{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  {εj \sp 
2}}+{2 \  {i \sp {2}} \  {j \sp {2}} \  εijk \  εik \  εj}+{{i \sp {2}} \  
{{j \sp {2}} \sp 2} \  ε1 \  {εik \sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  ε1 \  
{εijk \sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  {εij \sp 
2}} -{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  {k \sp {2}} \  ε1 \  {εi \sp 
2}}+{{{i \sp {2}} \sp 2} \  {{j \sp {2}} \sp 2} \  {k \sp {2}} \  {ε1 \sp 
3}}} & {{{i \sp {2}} \  {j \sp {2}} \  εijk \  {εk \sp 2}} -{2 \  {i \sp {2}} 
\  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εij \  εk} -{{i \sp {2}} \  εijk \  
{εjk \sp 2}} -{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εi \  
εjk}+{{i \sp {2}} \  {k \sp {2}} \  εijk \  {εj \sp 2}}+{2 \  {i \sp {2}} \  
{j \sp {2}} \  {k \sp {2}} \  ε1 \  εik \  εj} -{{j \sp {2}} \  εijk \  {εik 
\sp 2}}+{εijk \sp 3}+{{\left( -{{k \sp {2}} \  {εij \sp 2}}+{{j \sp {2}} \  
{k \sp {2}} \  {εi \sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {ε1 
\sp 2}} 
\right)}
\  εijk}} & {{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  εi \  {εk \sp 2}}+{2 \  
{i \sp {2}} \  {j \sp {2}} \  εij \  εjk \  εk}+{{i \sp {2}} \  {j \sp {2}} \  
εi \  {εjk \sp 2}}+{{\left( -{2 \  {i \sp {2}} \  {j \sp {2}} \  εik \  εj} 
-{2 \  {i \sp {2}} \  {j \sp {2}} \  ε1 \  εijk} 
\right)}
\  εjk}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εi \  {εj \sp 2}} -{{{j 
\sp {2}} \sp 2} \  εi \  {εik \sp 2}}+{{j \sp {2}} \  εi \  {εijk \sp 2}} 
-{{j \sp {2}} \  {k \sp {2}} \  εi \  {εij \sp 2}}+{{{j \sp {2}} \sp 2} \  {k 
\sp {2}} \  {εi \sp 3}} -{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  {k \sp {2}} \  
{ε1 \sp 2} \  εi}} & {{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  εj \  {εk \sp 
2}} -{2 \  {i \sp {2}} \  {j \sp {2}} \  εij \  εik \  εk} -{{{i \sp {2}} \sp 
2} \  εj \  {εjk \sp 2}} -{2 \  {i \sp {2}} \  {j \sp {2}} \  εi \  εik \  
εjk}+{{{i \sp {2}} \sp 2} \  {k \sp {2}} \  {εj \sp 3}}+{{\left( {{i \sp {2}} 
\  {j \sp {2}} \  {εik \sp 2}}+{{i \sp {2}} \  {εijk \sp 2}} -{{i \sp {2}} \  
{k \sp {2}} \  {εij \sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {εi 
\sp 2}} -{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  {k \sp {2}} \  {ε1 \sp 2}} 
\right)}
\  εj}+{2 \  {i \sp {2}} \  {j \sp {2}} \  ε1 \  εijk \  εik}} & {{{i \sp 
{2}} \  {j \sp {2}} \  εij \  {εk \sp 2}}+{{\left( {2 \  {i \sp {2}} \  {j 
\sp {2}} \  εi \  εjk} -{2 \  {i \sp {2}} \  {j \sp {2}} \  εik \  εj} -{2 \  
{i \sp {2}} \  {j \sp {2}} \  ε1 \  εijk} 
\right)}
\  εk}+{{i \sp {2}} \  εij \  {εjk \sp 2}} -{{i \sp {2}} \  {k \sp {2}} \  
εij \  {εj \sp 2}}+{{j \sp {2}} \  εij \  {εik \sp 2}} -{εij \  {εijk \sp 
2}}+{{k \sp {2}} \  {εij \sp 3}}+{{\left( -{{j \sp {2}} \  {k \sp {2}} \  {εi 
\sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {ε1 \sp 2}} 
\right)}
\  εij}} \ 
{{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εij \  {εk \sp 2}}+{{\left( {2 
\  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εi \  εjk} -{2 \  {i \sp {2}} 
\  {j \sp {2}} \  {k \sp {2}} \  εik \  εj} -{2 \  {i \sp {2}} \  {j \sp {2}} 
\  {k \sp {2}} \  ε1 \  εijk} 
\right)}
\  εk}+{{i \sp {2}} \  {k \sp {2}} \  εij \  {εjk \sp 2}} -{{i \sp {2}} \  
{{k \sp {2}} \sp 2} \  εij \  {εj \sp 2}}+{{j \sp {2}} \  {k \sp {2}} \  εij 
\  {εik \sp 2}} -{{k \sp {2}} \  εij \  {εijk \sp 2}}+{{{k \sp {2}} \sp 2} \  
{εij \sp 3}}+{{\left( -{{j \sp {2}} \  {{k \sp {2}} \sp 2} \  {εi \sp 2}}+{{i 
\sp {2}} \  {j \sp {2}} \  {{k \sp {2}} \sp 2} \  {ε1 \sp 2}} 
\right)}
\  εij}} & {{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εj \  {εk \sp 2}} 
-{2 \  {j \sp {2}} \  {k \sp {2}} \  εij \  εik \  εk} -{{i \sp {2}} \  {k 
\sp {2}} \  εj \  {εjk \sp 2}} -{2 \  {j \sp {2}} \  {k \sp {2}} \  εi \  εik 
\  εjk}+{{i \sp {2}} \  {{k \sp {2}} \sp 2} \  {εj \sp 3}}+{{\left( {{j \sp 
{2}} \  {k \sp {2}} \  {εik \sp 2}}+{{k \sp {2}} \  {εijk \sp 2}} -{{{k \sp 
{2}} \sp 2} \  {εij \sp 2}}+{{j \sp {2}} \  {{k \sp {2}} \sp 2} \  {εi \sp 
2}} -{{i \sp {2}} \  {j \sp {2}} \  {{k \sp {2}} \sp 2} \  {ε1 \sp 2}} 
\right)}
\  εj}+{2 \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εijk \  εik}} & {-{{i \sp 
{2}} \  {j \sp {2}} \  {k \sp {2}} \  εi \  {εk \sp 2}} -{2 \  {i \sp {2}} \  
{k \sp {2}} \  εij \  εjk \  εk} -{{i \sp {2}} \  {k \sp {2}} \  εi \  {εjk 
\sp 2}}+{{\left( {2 \  {i \sp {2}} \  {k \sp {2}} \  εik \  εj}+{2 \  {i \sp 
{2}} \  {k \sp {2}} \  ε1 \  εijk} 
\right)}
\  εjk} -{{i \sp {2}} \  {{k \sp {2}} \sp 2} \  εi \  {εj \sp 2}}+{{j \sp 
{2}} \  {k \sp {2}} \  εi \  {εik \sp 2}} -{{k \sp {2}} \  εi \  {εijk \sp 
2}}+{{{k \sp {2}} \sp 2} \  εi \  {εij \sp 2}} -{{j \sp {2}} \  {{k \sp {2}} 
\sp 2} \  {εi \sp 3}}+{{i \sp {2}} \  {j \sp {2}} \  {{k \sp {2}} \sp 2} \  
{ε1 \sp 2} \  εi}} & {{{i \sp {2}} \  {j \sp {2}} \  εijk \  {εk \sp 2}} -{2 
\  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εij \  εk} -{{i \sp 
{2}} \  εijk \  {εjk \sp 2}} -{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} 
\  ε1 \  εi \  εjk}+{{i \sp {2}} \  {k \sp {2}} \  εijk \  {εj \sp 2}}+{2 \  
{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εik \  εj} -{{j \sp {2}} \  
εijk \  {εik \sp 2}}+{εijk \sp 3}+{{\left( -{{k \sp {2}} \  {εij \sp 2}}+{{j 
\sp {2}} \  {k \sp {2}} \  {εi \sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp 
{2}} \  {ε1 \sp 2}} 
\right)}
\  εijk}} & {{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  {εk \sp 
2}}+{2 \  {k \sp {2}} \  εij \  εijk \  εk} -{{i \sp {2}} \  {k \sp {2}} \  
ε1 \  {εjk \sp 2}}+{2 \  {k \sp {2}} \  εi \  εijk \  εjk}+{{i \sp {2}} \  
{{k \sp {2}} \sp 2} \  ε1 \  {εj \sp 2}} -{2 \  {k \sp {2}} \  εijk \  εik \  
εj} -{{j \sp {2}} \  {k \sp {2}} \  ε1 \  {εik \sp 2}} -{{k \sp {2}} \  ε1 \  
{εijk \sp 2}} -{{{k \sp {2}} \sp 2} \  ε1 \  {εij \sp 2}}+{{j \sp {2}} \  {{k 
\sp {2}} \sp 2} \  ε1 \  {εi \sp 2}} -{{i \sp {2}} \  {j \sp {2}} \  {{k \sp 
{2}} \sp 2} \  {ε1 \sp 3}}} & {-{{i \sp {2}} \  {j \sp {2}} \  εjk \  {εk \sp 
2}}+{2 \  {j \sp {2}} \  {k \sp {2}} \  εi \  εij \  εk}+{{i \sp {2}} \  {εjk 
\sp 3}}+{{\left( -{{i \sp {2}} \  {k \sp {2}} \  {εj \sp 2}}+{{j \sp {2}} \  
{εik \sp 2}} -{εijk \sp 2}+{{k \sp {2}} \  {εij \sp 2}}+{{j \sp {2}} \  {k 
\sp {2}} \  {εi \sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {ε1 \sp 
2}} 
\right)}
\  εjk} -{2 \  {j \sp {2}} \  {k \sp {2}} \  εi \  εik \  εj} -{2 \  {j \sp 
{2}} \  {k \sp {2}} \  ε1 \  εi \  εijk}} & {{{i \sp {2}} \  {j \sp {2}} \  
εik \  {εk \sp 2}}+{2 \  {i \sp {2}} \  {k \sp {2}} \  εij \  εj \  εk} -{{i 
\sp {2}} \  εik \  {εjk \sp 2}}+{2 \  {i \sp {2}} \  {k \sp {2}} \  εi \  εj 
\  εjk} -{{i \sp {2}} \  {k \sp {2}} \  εik \  {εj \sp 2}} -{2 \  {i \sp {2}} 
\  {k \sp {2}} \  ε1 \  εijk \  εj} -{{j \sp {2}} \  {εik \sp 3}}+{{\left( 
{εijk \sp 2} -{{k \sp {2}} \  {εij \sp 2}}+{{j \sp {2}} \  {k \sp {2}} \  {εi 
\sp 2}} -{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {ε1 \sp 2}} 
\right)}
\  εik}} & {-{{i \sp {2}} \  {j \sp {2}} \  {εk \sp 3}}+{{\left( {{i \sp {2}} 
\  {εjk \sp 2}} -{{i \sp {2}} \  {k \sp {2}} \  {εj \sp 2}}+{{j \sp {2}} \  
{εik \sp 2}} -{εijk \sp 2} -{{k \sp {2}} \  {εij \sp 2}} -{{j \sp {2}} \  {k 
\sp {2}} \  {εi \sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {ε1 \sp 
2}} 
\right)}
\  εk} -{2 \  {k \sp {2}} \  εi \  εij \  εjk}+{2 \  {k \sp {2}} \  εij \  
εik \  εj}+{2 \  {k \sp {2}} \  ε1 \  εij \  εijk}} \ 
{-{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  εik \  {εk \sp 2}} -{2 \  {i \sp 
{2}} \  {j \sp {2}} \  {k \sp {2}} \  εij \  εj \  εk}+{{i \sp {2}} \  {j \sp 
{2}} \  εik \  {εjk \sp 2}} -{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} 
\  εi \  εj \  εjk}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εik \  {εj 
\sp 2}}+{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εijk \  
εj}+{{{j \sp {2}} \sp 2} \  {εik \sp 3}}+{{\left( -{{j \sp {2}} \  {εijk \sp 
2}}+{{j \sp {2}} \  {k \sp {2}} \  {εij \sp 2}} -{{{j \sp {2}} \sp 2} \  {k 
\sp {2}} \  {εi \sp 2}}+{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  {k \sp {2}} \  
{ε1 \sp 2}} 
\right)}
\  εik}} & {{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  {εk \sp 3}}+{{\left( -{{i 
\sp {2}} \  {j \sp {2}} \  {εjk \sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp 
{2}} \  {εj \sp 2}} -{{{j \sp {2}} \sp 2} \  {εik \sp 2}}+{{j \sp {2}} \  
{εijk \sp 2}}+{{j \sp {2}} \  {k \sp {2}} \  {εij \sp 2}}+{{{j \sp {2}} \sp 
2} \  {k \sp {2}} \  {εi \sp 2}} -{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  {k 
\sp {2}} \  {ε1 \sp 2}} 
\right)}
\  εk}+{2 \  {j \sp {2}} \  {k \sp {2}} \  εi \  εij \  εjk} -{2 \  {j \sp 
{2}} \  {k \sp {2}} \  εij \  εik \  εj} -{2 \  {j \sp {2}} \  {k \sp {2}} \  
ε1 \  εij \  εijk}} & {-{{i \sp {2}} \  {j \sp {2}} \  εijk \  {εk \sp 2}}+{2 
\  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εij \  εk}+{{i \sp {2}} 
\  εijk \  {εjk \sp 2}}+{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 
\  εi \  εjk} -{{i \sp {2}} \  {k \sp {2}} \  εijk \  {εj \sp 2}} -{2 \  {i 
\sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εik \  εj}+{{j \sp {2}} \  
εijk \  {εik \sp 2}} -{εijk \sp 3}+{{\left( {{k \sp {2}} \  {εij \sp 2}} -{{j 
\sp {2}} \  {k \sp {2}} \  {εi \sp 2}} -{{i \sp {2}} \  {j \sp {2}} \  {k \sp 
{2}} \  {ε1 \sp 2}} 
\right)}
\  εijk}} & {-{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  εi \  {εk \sp 2}} -{2 \  
{i \sp {2}} \  {j \sp {2}} \  εij \  εjk \  εk} -{{i \sp {2}} \  {j \sp {2}} 
\  εi \  {εjk \sp 2}}+{{\left( {2 \  {i \sp {2}} \  {j \sp {2}} \  εik \  
εj}+{2 \  {i \sp {2}} \  {j \sp {2}} \  ε1 \  εijk} 
\right)}
\  εjk} -{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εi \  {εj \sp 2}}+{{{j 
\sp {2}} \sp 2} \  εi \  {εik \sp 2}} -{{j \sp {2}} \  εi \  {εijk \sp 
2}}+{{j \sp {2}} \  {k \sp {2}} \  εi \  {εij \sp 2}} -{{{j \sp {2}} \sp 2} \  
{k \sp {2}} \  {εi \sp 3}}+{{i \sp {2}} \  {{j \sp {2}} \sp 2} \  {k \sp {2}} 
\  {ε1 \sp 2} \  εi}} & {{{i \sp {2}} \  {j \sp {2}} \  εjk \  {εk \sp 2}} 
-{2 \  {j \sp {2}} \  {k \sp {2}} \  εi \  εij \  εk} -{{i \sp {2}} \  {εjk 
\sp 3}}+{{\left( {{i \sp {2}} \  {k \sp {2}} \  {εj \sp 2}} -{{j \sp {2}} \  
{εik \sp 2}}+{εijk \sp 2} -{{k \sp {2}} \  {εij \sp 2}} -{{j \sp {2}} \  {k 
\sp {2}} \  {εi \sp 2}} -{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {ε1 
\sp 2}} 
\right)}
\  εjk}+{2 \  {j \sp {2}} \  {k \sp {2}} \  εi \  εik \  εj}+{2 \  {j \sp 
{2}} \  {k \sp {2}} \  ε1 \  εi \  εijk}} & {{{i \sp {2}} \  {{j \sp {2}} \sp 
2} \  ε1 \  {εk \sp 2}}+{2 \  {j \sp {2}} \  εij \  εijk \  εk} -{{i \sp {2}} 
\  {j \sp {2}} \  ε1 \  {εjk \sp 2}}+{2 \  {j \sp {2}} \  εi \  εijk \  
εjk}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  {εj \sp 2}} -{2 \  
{j \sp {2}} \  εijk \  εik \  εj} -{{{j \sp {2}} \sp 2} \  ε1 \  {εik \sp 2}} 
-{{j \sp {2}} \  ε1 \  {εijk \sp 2}} -{{j \sp {2}} \  {k \sp {2}} \  ε1 \  
{εij \sp 2}}+{{{j \sp {2}} \sp 2} \  {k \sp {2}} \  ε1 \  {εi \sp 2}} -{{i 
\sp {2}} \  {{j \sp {2}} \sp 2} \  {k \sp {2}} \  {ε1 \sp 3}}} & {{{i \sp 
{2}} \  {j \sp {2}} \  εij \  {εk \sp 2}}+{{\left( {2 \  {i \sp {2}} \  {j 
\sp {2}} \  εi \  εjk} -{2 \  {i \sp {2}} \  {j \sp {2}} \  εik \  εj} -{2 \  
{i \sp {2}} \  {j \sp {2}} \  ε1 \  εijk} 
\right)}
\  εk}+{{i \sp {2}} \  εij \  {εjk \sp 2}} -{{i \sp {2}} \  {k \sp {2}} \  
εij \  {εj \sp 2}}+{{j \sp {2}} \  εij \  {εik \sp 2}} -{εij \  {εijk \sp 
2}}+{{k \sp {2}} \  {εij \sp 3}}+{{\left( -{{j \sp {2}} \  {k \sp {2}} \  {εi 
\sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {ε1 \sp 2}} 
\right)}
\  εij}} & {{{i \sp {2}} \  {j \sp {2}} \  εj \  {εk \sp 2}} -{2 \  {j \sp 
{2}} \  εij \  εik \  εk} -{{i \sp {2}} \  εj \  {εjk \sp 2}} -{2 \  {j \sp 
{2}} \  εi \  εik \  εjk}+{{i \sp {2}} \  {k \sp {2}} \  {εj \sp 3}}+{{\left( 
{{j \sp {2}} \  {εik \sp 2}}+{εijk \sp 2} -{{k \sp {2}} \  {εij \sp 2}}+{{j 
\sp {2}} \  {k \sp {2}} \  {εi \sp 2}} -{{i \sp {2}} \  {j \sp {2}} \  {k \sp 
{2}} \  {ε1 \sp 2}} 
\right)}
\  εj}+{2 \  {j \sp {2}} \  ε1 \  εijk \  εik}} \ 
{-{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  εjk \  {εk \sp 2}}+{2 \  {i \sp {2}} 
\  {j \sp {2}} \  {k \sp {2}} \  εi \  εij \  εk}+{{{i \sp {2}} \sp 2} \  
{εjk \sp 3}}+{{\left( -{{{i \sp {2}} \sp 2} \  {k \sp {2}} \  {εj \sp 2}}+{{i 
\sp {2}} \  {j \sp {2}} \  {εik \sp 2}} -{{i \sp {2}} \  {εijk \sp 2}}+{{i 
\sp {2}} \  {k \sp {2}} \  {εij \sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp 
{2}} \  {εi \sp 2}}+{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  {k \sp {2}} \  {ε1 
\sp 2}} 
\right)}
\  εjk} -{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  εi \  εik \  εj} 
-{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εi \  εijk}} & {{{i 
\sp {2}} \  {j \sp {2}} \  εijk \  {εk \sp 2}} -{2 \  {i \sp {2}} \  {j \sp 
{2}} \  {k \sp {2}} \  ε1 \  εij \  εk} -{{i \sp {2}} \  εijk \  {εjk \sp 2}} 
-{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εi \  εjk}+{{i \sp 
{2}} \  {k \sp {2}} \  εijk \  {εj \sp 2}}+{2 \  {i \sp {2}} \  {j \sp {2}} \  
{k \sp {2}} \  ε1 \  εik \  εj} -{{j \sp {2}} \  εijk \  {εik \sp 2}}+{εijk 
\sp 3}+{{\left( -{{k \sp {2}} \  {εij \sp 2}}+{{j \sp {2}} \  {k \sp {2}} \  
{εi \sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {ε1 \sp 2}} 
\right)}
\  εijk}} & {{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  {εk \sp 3}}+{{\left( 
-{{{i \sp {2}} \sp 2} \  {εjk \sp 2}}+{{{i \sp {2}} \sp 2} \  {k \sp {2}} \  
{εj \sp 2}} -{{i \sp {2}} \  {j \sp {2}} \  {εik \sp 2}}+{{i \sp {2}} \  
{εijk \sp 2}}+{{i \sp {2}} \  {k \sp {2}} \  {εij \sp 2}}+{{i \sp {2}} \  {j 
\sp {2}} \  {k \sp {2}} \  {εi \sp 2}} -{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  
{k \sp {2}} \  {ε1 \sp 2}} 
\right)}
\  εk}+{2 \  {i \sp {2}} \  {k \sp {2}} \  εi \  εij \  εjk} -{2 \  {i \sp 
{2}} \  {k \sp {2}} \  εij \  εik \  εj} -{2 \  {i \sp {2}} \  {k \sp {2}} \  
ε1 \  εij \  εijk}} & {-{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  εj \  {εk \sp 
2}}+{2 \  {i \sp {2}} \  {j \sp {2}} \  εij \  εik \  εk}+{{{i \sp {2}} \sp 
2} \  εj \  {εjk \sp 2}}+{2 \  {i \sp {2}} \  {j \sp {2}} \  εi \  εik \  
εjk} -{{{i \sp {2}} \sp 2} \  {k \sp {2}} \  {εj \sp 3}}+{{\left( -{{i \sp 
{2}} \  {j \sp {2}} \  {εik \sp 2}} -{{i \sp {2}} \  {εijk \sp 2}}+{{i \sp 
{2}} \  {k \sp {2}} \  {εij \sp 2}} -{{i \sp {2}} \  {j \sp {2}} \  {k \sp 
{2}} \  {εi \sp 2}}+{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  {k \sp {2}} \  {ε1 
\sp 2}} 
\right)}
\  εj} -{2 \  {i \sp {2}} \  {j \sp {2}} \  ε1 \  εijk \  εik}} & {-{{i \sp 
{2}} \  {j \sp {2}} \  εik \  {εk \sp 2}} -{2 \  {i \sp {2}} \  {k \sp {2}} \  
εij \  εj \  εk}+{{i \sp {2}} \  εik \  {εjk \sp 2}} -{2 \  {i \sp {2}} \  {k 
\sp {2}} \  εi \  εj \  εjk}+{{i \sp {2}} \  {k \sp {2}} \  εik \  {εj \sp 
2}}+{2 \  {i \sp {2}} \  {k \sp {2}} \  ε1 \  εijk \  εj}+{{j \sp {2}} \  
{εik \sp 3}}+{{\left( -{εijk \sp 2}+{{k \sp {2}} \  {εij \sp 2}} -{{j \sp 
{2}} \  {k \sp {2}} \  {εi \sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} 
\  {ε1 \sp 2}} 
\right)}
\  εik}} & {-{{i \sp {2}} \  {j \sp {2}} \  εij \  {εk \sp 2}}+{{\left( -{2 \  
{i \sp {2}} \  {j \sp {2}} \  εi \  εjk}+{2 \  {i \sp {2}} \  {j \sp {2}} \  
εik \  εj}+{2 \  {i \sp {2}} \  {j \sp {2}} \  ε1 \  εijk} 
\right)}
\  εk} -{{i \sp {2}} \  εij \  {εjk \sp 2}}+{{i \sp {2}} \  {k \sp {2}} \  
εij \  {εj \sp 2}} -{{j \sp {2}} \  εij \  {εik \sp 2}}+{εij \  {εijk \sp 2}} 
-{{k \sp {2}} \  {εij \sp 3}}+{{\left( {{j \sp {2}} \  {k \sp {2}} \  {εi \sp 
2}} -{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {ε1 \sp 2}} 
\right)}
\  εij}} & {{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  ε1 \  {εk \sp 2}}+{2 \  {i 
\sp {2}} \  εij \  εijk \  εk} -{{{i \sp {2}} \sp 2} \  ε1 \  {εjk \sp 2}}+{2 
\  {i \sp {2}} \  εi \  εijk \  εjk}+{{{i \sp {2}} \sp 2} \  {k \sp {2}} \  
ε1 \  {εj \sp 2}} -{2 \  {i \sp {2}} \  εijk \  εik \  εj} -{{i \sp {2}} \  
{j \sp {2}} \  ε1 \  {εik \sp 2}} -{{i \sp {2}} \  ε1 \  {εijk \sp 2}} -{{i 
\sp {2}} \  {k \sp {2}} \  ε1 \  {εij \sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  
{k \sp {2}} \  ε1 \  {εi \sp 2}} -{{{i \sp {2}} \sp 2} \  {j \sp {2}} \  {k 
\sp {2}} \  {ε1 \sp 3}}} & {-{{i \sp {2}} \  {j \sp {2}} \  εi \  {εk \sp 2}} 
-{2 \  {i \sp {2}} \  εij \  εjk \  εk} -{{i \sp {2}} \  εi \  {εjk \sp 
2}}+{{\left( {2 \  {i \sp {2}} \  εik \  εj}+{2 \  {i \sp {2}} \  ε1 \  εijk} 
\right)}
\  εjk} -{{i \sp {2}} \  {k \sp {2}} \  εi \  {εj \sp 2}}+{{j \sp {2}} \  εi 
\  {εik \sp 2}} -{εi \  {εijk \sp 2}}+{{k \sp {2}} \  εi \  {εij \sp 2}} -{{j 
\sp {2}} \  {k \sp {2}} \  {εi \sp 3}}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp 
{2}} \  {ε1 \sp 2} \  εi}} \ 
{{{i \sp {2}} \  {j \sp {2}} \  εijk \  {εk \sp 2}} -{2 \  {i \sp {2}} \  {j 
\sp {2}} \  {k \sp {2}} \  ε1 \  εij \  εk} -{{i \sp {2}} \  εijk \  {εjk \sp 
2}} -{2 \  {i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  ε1 \  εi \  εjk}+{{i 
\sp {2}} \  {k \sp {2}} \  εijk \  {εj \sp 2}}+{2 \  {i \sp {2}} \  {j \sp 
{2}} \  {k \sp {2}} \  ε1 \  εik \  εj} -{{j \sp {2}} \  εijk \  {εik \sp 
2}}+{εijk \sp 3}+{{\left( -{{k \sp {2}} \  {εij \sp 2}}+{{j \sp {2}} \  {k 
\sp {2}} \  {εi \sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {ε1 \sp 
2}} 
\right)}
\  εijk}} & {-{{i \sp {2}} \  {j \sp {2}} \  εjk \  {εk \sp 2}}+{2 \  {j \sp 
{2}} \  {k \sp {2}} \  εi \  εij \  εk}+{{i \sp {2}} \  {εjk \sp 3}}+{{\left( 
-{{i \sp {2}} \  {k \sp {2}} \  {εj \sp 2}}+{{j \sp {2}} \  {εik \sp 2}} 
-{εijk \sp 2}+{{k \sp {2}} \  {εij \sp 2}}+{{j \sp {2}} \  {k \sp {2}} \  {εi 
\sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {ε1 \sp 2}} 
\right)}
\  εjk} -{2 \  {j \sp {2}} \  {k \sp {2}} \  εi \  εik \  εj} -{2 \  {j \sp 
{2}} \  {k \sp {2}} \  ε1 \  εi \  εijk}} & {{{i \sp {2}} \  {j \sp {2}} \  
εik \  {εk \sp 2}}+{2 \  {i \sp {2}} \  {k \sp {2}} \  εij \  εj \  εk} -{{i 
\sp {2}} \  εik \  {εjk \sp 2}}+{2 \  {i \sp {2}} \  {k \sp {2}} \  εi \  εj 
\  εjk} -{{i \sp {2}} \  {k \sp {2}} \  εik \  {εj \sp 2}} -{2 \  {i \sp {2}} 
\  {k \sp {2}} \  ε1 \  εijk \  εj} -{{j \sp {2}} \  {εik \sp 3}}+{{\left( 
{εijk \sp 2} -{{k \sp {2}} \  {εij \sp 2}}+{{j \sp {2}} \  {k \sp {2}} \  {εi 
\sp 2}} -{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {ε1 \sp 2}} 
\right)}
\  εik}} & {{{i \sp {2}} \  {j \sp {2}} \  εij \  {εk \sp 2}}+{{\left( {2 \  
{i \sp {2}} \  {j \sp {2}} \  εi \  εjk} -{2 \  {i \sp {2}} \  {j \sp {2}} \  
εik \  εj} -{2 \  {i \sp {2}} \  {j \sp {2}} \  ε1 \  εijk} 
\right)}
\  εk}+{{i \sp {2}} \  εij \  {εjk \sp 2}} -{{i \sp {2}} \  {k \sp {2}} \  
εij \  {εj \sp 2}}+{{j \sp {2}} \  εij \  {εik \sp 2}} -{εij \  {εijk \sp 
2}}+{{k \sp {2}} \  {εij \sp 3}}+{{\left( -{{j \sp {2}} \  {k \sp {2}} \  {εi 
\sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {ε1 \sp 2}} 
\right)}
\  εij}} & {-{{i \sp {2}} \  {j \sp {2}} \  {εk \sp 3}}+{{\left( {{i \sp {2}} 
\  {εjk \sp 2}} -{{i \sp {2}} \  {k \sp {2}} \  {εj \sp 2}}+{{j \sp {2}} \  
{εik \sp 2}} -{εijk \sp 2} -{{k \sp {2}} \  {εij \sp 2}} -{{j \sp {2}} \  {k 
\sp {2}} \  {εi \sp 2}}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {ε1 \sp 
2}} 
\right)}
\  εk} -{2 \  {k \sp {2}} \  εi \  εij \  εjk}+{2 \  {k \sp {2}} \  εij \  
εik \  εj}+{2 \  {k \sp {2}} \  ε1 \  εij \  εijk}} & {{{i \sp {2}} \  {j \sp 
{2}} \  εj \  {εk \sp 2}} -{2 \  {j \sp {2}} \  εij \  εik \  εk} -{{i \sp 
{2}} \  εj \  {εjk \sp 2}} -{2 \  {j \sp {2}} \  εi \  εik \  εjk}+{{i \sp 
{2}} \  {k \sp {2}} \  {εj \sp 3}}+{{\left( {{j \sp {2}} \  {εik \sp 
2}}+{εijk \sp 2} -{{k \sp {2}} \  {εij \sp 2}}+{{j \sp {2}} \  {k \sp {2}} \  
{εi \sp 2}} -{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {ε1 \sp 2}} 
\right)}
\  εj}+{2 \  {j \sp {2}} \  ε1 \  εijk \  εik}} & {-{{i \sp {2}} \  {j \sp 
{2}} \  εi \  {εk \sp 2}} -{2 \  {i \sp {2}} \  εij \  εjk \  εk} -{{i \sp 
{2}} \  εi \  {εjk \sp 2}}+{{\left( {2 \  {i \sp {2}} \  εik \  εj}+{2 \  {i 
\sp {2}} \  ε1 \  εijk} 
\right)}
\  εjk} -{{i \sp {2}} \  {k \sp {2}} \  εi \  {εj \sp 2}}+{{j \sp {2}} \  εi 
\  {εik \sp 2}} -{εi \  {εijk \sp 2}}+{{k \sp {2}} \  εi \  {εij \sp 2}} -{{j 
\sp {2}} \  {k \sp {2}} \  {εi \sp 3}}+{{i \sp {2}} \  {j \sp {2}} \  {k \sp 
{2}} \  {ε1 \sp 2} \  εi}} & {{{i \sp {2}} \  {j \sp {2}} \  ε1 \  {εk \sp 
2}}+{2 \  εij \  εijk \  εk} -{{i \sp {2}} \  ε1 \  {εjk \sp 2}}+{2 \  εi \  
εijk \  εjk}+{{i \sp {2}} \  {k \sp {2}} \  ε1 \  {εj \sp 2}} -{2 \  εijk \  
εik \  εj} -{{j \sp {2}} \  ε1 \  {εik \sp 2}} -{ε1 \  {εijk \sp 2}} -{{k \sp 
{2}} \  ε1 \  {εij \sp 2}}+{{j \sp {2}} \  {k \sp {2}} \  ε1 \  {εi \sp 2}} 
-{{i \sp {2}} \  {j \sp {2}} \  {k \sp {2}} \  {ε1 \sp 3}}} 
\end{array}
\right]
\leqno(54)
$$
Type: Matrix(SparseMultivariatePolynomial(Integer,Kernel(Expression(Integer))))
(55) -> O:𝐋:= Ω / Ų
$$
8 
\leqno(55)
$$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
test
    (    I ΩX     )  /
    (     Ų I     )  =  I
$$
true 
\leqno(56)
$$
                                                                Type: Boolean

test
    (     ΩX I    )  /
    (    I Ų      )  =  I
$$
true 
\leqno(57)
$$
                                                                Type: Boolean
(58) -> eval(Ω,ck)

$$
{{1 \over 8} \  {| \sb {{ \  1 \  1}}}}+{{1 \over {8 \  {i \sp {2}}}} \  {| 
\sb {{ \  i \  i}}}}+{{1 \over {8 \  {j \sp {2}}}} \  {| \sb {{ \  j \  
j}}}}+{{1 \over {8 \  {k \sp {2}}}} \  {| \sb {{ \  k \  k}}}} -{{1 \over {8 
\  {i \sp {2}} \  {j \sp {2}}}} \  {| \sb {{ \  ij \  ij}}}} -{{1 \over {8 \  
{i \sp {2}} \  {k \sp {2}}}} \  {| \sb {{ \  ik \  ik}}}} -{{1 \over {8 \  {j 
\sp {2}} \  {k \sp {2}}}} \  {| \sb {{ \  jk \  jk}}}} -{{1 \over {8 \  {i 
\sp {2}} \  {j \sp {2}} \  {k \sp {2}}}} \  {| \sb {{ \  ijk \  ijk}}}} 
\leqno(58)
$$
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
(59) -> W:= (Y I) / Ų;
  (  ΩX I ΩX  ) /
  (  I  W  I  )

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l.169 a:=Σ(sb('a,[i])*𝐞.i, i,1..dim)

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