Edit detail for SandBox Quaternion Algebra is Frobenius in Many Ways revision 22 of 32

Quaternion Algebra Is Frobenius In Many Ways

Linear operators over a 4-dimensional vector space representing quaternion 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


We need the Axiom LinearOperator library.
fricas
(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/aw/var/LatexWiki/CARTEN.NRLIB/CARTEN
   Arity is now explicitly exposed in frame initial 
   Arity will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/ARITY.NRLIB/ARITY
   ClosedMonoidal is now explicitly exposed in frame initial 
   ClosedMonoidal will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CMONAL.NRLIB/CMONAL
   ClosedProp is now explicitly exposed in frame initial 
   ClosedProp will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CPROP.NRLIB/CPROP
   ClosedLinearOperator is now explicitly exposed in frame initial 
   ClosedLinearOperator will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CLOP.NRLIB/CLOP
   CaleyDickson is now explicitly exposed in frame initial 
   CaleyDickson will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CALEY.NRLIB/CALEY

Use the following macros for convenient notation
fricas
-- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])
Type: Void
fricas
-- list
macro Ξ(f,i,n)==[f for i in n]
Type: Void
fricas
-- 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.
fricas
dim:=4
\begin{equation}
\label{eq1}4\end{equation}
Type: PositiveInteger?
fricas
macro ℒ == List
Type: Void
fricas
macro ℂ == CaleyDickson
Type: Void
fricas
macro ℚ == Expression Integer
Type: Void
fricas
𝐋 := ClosedLinearOperator(OVAR ['1,'i,'j,'k], ℚ)
\begin{equation*}
\label{eq2}\hbox{\axiomType{ClosedLinearOperator?}\ } \left({{\hbox{\axiomType{OrderedVariableList?}\ } \left({\left[ 1, \: i , \: j , \: k \right]?}\right)}, \:{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}}\right)\end{equation*}
Type: Type
fricas
𝐞:ℒ 𝐋      := basisOut()
\begin{equation*}
\label{eq3}\left[{|_{\  1}}, \:{|_{\  i}}, \:{|_{\  j}}, \:{|_{\  k}}\right]?\end{equation*}
Type: List(ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer)))
fricas
𝐝:ℒ 𝐋      := basisIn()
\begin{equation*}
\label{eq4}\left[{|^{\  1}}, \:{|^{\  i}}, \:{|^{\  j}}, \:{|^{\  k}}\right]?\end{equation*}
Type: List(ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer)))
fricas
I:𝐋:=[1]   -- identity for composition
\begin{equation}
\label{eq5}{|_{\  1}^{\  1}}+{|_{\  i}^{\  i}}+{|_{\  j}^{\  j}}+{|_{\  k}^{\  k}}\end{equation}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
X:𝐋:=[2,1] -- twist
\begin{equation}
\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}}
\end{array}
\end{equation}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
V:𝐋:=ev(1) -- evaluation
\begin{equation}
\label{eq7}{|^{\  1 \  1}}+{|^{\  i \  i}}+{|^{\  j \  j}}+{|^{\  k \  k}}\end{equation}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
Λ:𝐋:=co(1) -- co-evaluation
\begin{equation}
\label{eq8}{|_{\  1 \  1}}+{|_{\  i \  i}}+{|_{\  j \  j}}+{|_{\  k \  k}}\end{equation}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
equate(eq)==map((x,y)+->(x=y),ravel lhs eq, ravel rhs eq);
Type: Void

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)
fricas
--q0:=sb('q,[0])
q0:=σ::Symbol
\begin{equation*}
\label{eq9}��\end{equation*}
Type: Symbol
fricas
--q0:=1  -- not split-complex
--q1:=sb('q,[1])
q1:=μ::Symbol
\begin{equation*}
\label{eq10}��\end{equation*}
Type: Symbol
fricas
--q1:=1  -- quaternion
QQ := ℂ(ℂ(ℚ,'i,q0),'j,q1);
Type: Type

Basis: Each B.i is a quaternion number
fricas
B:ℒ QQ := map(x +-> hyper x,1$SQMATRIX(dim,ℚ)::ℒ ℒ ℚ)
\begin{equation*}
\label{eq11}\left[ 1, \: i , \: j , \:{ij}\right]?\end{equation*}
Type: List(CaleyDickson?(CaleyDickson?(Expression(Integer),i,σ),j,μ))
fricas
-- Multiplication table:
M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j, i,1..dim), j,1..dim)
\begin{equation*}
\label{eq12}\left[ 
\begin{array}{cccc}
1 & i & j &{ij}
\
i & - �� & -{ij}&{�� j}
\
j &{ij}& - �� &{- �� i}
\
{ij}&{- �� j}&{�� i}& -{�� \  ��}
\end{array}
\right]\end{equation*}
Type: Matrix(CaleyDickson?(CaleyDickson?(Expression(Integer),i,σ),j,μ))
fricas
-- Function to divide the matrix entries by a basis element
S(y) == map(x +-> real real(x/y),M)
Type: Void
fricas
-- The result is a nested list
ѕ :=map(S,B)::ℒ ℒ ℒ ℚ
fricas
Compiling function S with type CaleyDickson(CaleyDickson(Expression(
      Integer),i,σ),j,μ) -> Matrix(Expression(Integer))
\begin{equation*}
\label{eq13}\begin{array}{@{}l}
\displaystyle
\left[{\left[{\left[ 1, \: 0, \: 0, \: 0 \right]?}, \:{\left[ 0, \: - �� , \: 0, \: 0 \right]?}, \:{\left[ 0, \: 0, \: - �� , \: 0 \right]?}, \:{\left[ 0, \: 0, \: 0, \: -{�� \  ��}\right]?}\right]}, \: \right.
\
\
\displaystyle
\left.{
\begin{array}{@{}l}
\displaystyle
\left[{\left[ 0, \:{\frac{��}{\overline ��}}, \: 0, \: 0 \right]?}, \:{\left[{\frac{��}{\overline ��}}, \: 0, \: 0, \: 0 \right]?}, \:{\left[ 0, \: 0, \: 0, \: -{\frac{�� \  ��}{\overline ��}}\right]?}, \:{\left[ 0, \: 0, \:{\frac{�� \  ��}{\overline ��}}, \: 0 \right]?}\right] 
\end{array}
}, \: \right.
\
\
\displaystyle
\left.{
\begin{array}{@{}l}
\displaystyle
\left[{\left[ 0, \: 0, \:{\frac{��}{\overline ��}}, \: 0 \right]?}, \:{\left[ 0, \: 0, \: 0, \:{\frac{�� \  ��}{\overline ��}}\right]?}, \:{\left[{\frac{��}{\overline ��}}, \: 0, \: 0, \: 0 \right]?}, \:{\left[ 0, \: -{\frac{�� \  ��}{\overline ��}}, \: 0, \: 0 \right]?}\right] 
\end{array}
}, \:{\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] 
\end{array}
\end{equation*}
Type: List(List(List(Expression(Integer))))
fricas
-- 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)
\begin{equation*}
\label{eq14}\begin{array}{@{}l}
\displaystyle
{|_{\  1}^{\  1 \  1}}+{{\frac{��}{\overline ��}}\ {|_{\  i}^{\  1 \  i}}}+{{\frac{��}{\overline ��}}\ {|_{\  j}^{\  1 \  j}}}+{|_{\  k}^{\  1 \  k}}+{{\frac{��}{\overline ��}}\ {|_{\  i}^{\  i \  1}}}- 
\
\
\displaystyle
{�� \ {|_{\  1}^{\  i \  i}}}+{|_{\  k}^{\  i \  j}}-{{\frac{�� \  ��}{\overline ��}}\ {|_{\  j}^{\  i \  k}}}+{{\frac{��}{\overline ��}}\ {|_{\  j}^{\  j \  1}}}-{|_{\  k}^{\  j \  i}}- 
\
\
\displaystyle
{�� \ {|_{\  1}^{\  j \  j}}}+{{\frac{�� \  ��}{\overline ��}}\ {|_{\  i}^{\  j \  k}}}+{|_{\  k}^{\  k \  1}}+{{\frac{�� \  ��}{\overline ��}}\ {|_{\  j}^{\  k \  i}}}- 
\
\
\displaystyle
{{\frac{�� \  ��}{\overline ��}}\ {|_{\  i}^{\  k \  j}}}-{�� \  �� \ {|_{\  1}^{\  k \  k}}}
\end{array}
\end{equation*}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
arity Y
\begin{equation}
\label{eq15}\frac{{+}^{2}}{+}\end{equation}
Type: ClosedProp?(ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer)))
fricas
matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y, i,1..dim), j,1..dim)
\begin{equation*}
\label{eq16}\left[ 
\begin{array}{cccc}
{|_{\  1}}&{{\frac{��}{\overline ��}}\ {|_{\  i}}}&{{\frac{��}{\overline ��}}\ {|_{\  j}}}&{|_{\  k}}
\
{{\frac{��}{\overline ��}}\ {|_{\  i}}}& -{�� \ {|_{\  1}}}& -{|_{\  k}}&{{\frac{�� \  ��}{\overline ��}}\ {|_{\  j}}}
\
{{\frac{��}{\overline ��}}\ {|_{\  j}}}&{|_{\  k}}& -{�� \ {|_{\  1}}}& -{{\frac{�� \  ��}{\overline ��}}\ {|_{\  i}}}
\
{|_{\  k}}& -{{\frac{�� \  ��}{\overline ��}}\ {|_{\  j}}}&{{\frac{�� \  ��}{\overline ��}}\ {|_{\  i}}}& -{�� \  �� \ {|_{\  1}}}
\end{array}
\right]\end{equation*}
Type: Matrix(ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer)))

Multiplication of arbitrary quaternions $a$ and $b$
fricas
a:=Σ(sb('a,[i])*𝐞.i, i,1..dim)
\begin{equation}
\label{eq17}{{a_{1}}\ {|_{\  1}}}+{{a_{2}}\ {|_{\  i}}}+{{a_{3}}\ {|_{\  j}}}+{{a_{4}}\ {|_{\  k}}}\end{equation}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
b:=Σ(sb('b,[i])*𝐞.i, i,1..dim)
\begin{equation}
\label{eq18}{{b_{1}}\ {|_{\  1}}}+{{b_{2}}\ {|_{\  i}}}+{{b_{3}}\ {|_{\  j}}}+{{b_{4}}\ {|_{\  k}}}\end{equation}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
(a*b)/Y
\begin{equation*}
\label{eq19}\begin{array}{@{}l}
\displaystyle
{{\left({{\left(-{{a_{4}}\ {b_{4}}\  ��}-{{a_{2}}\ {b_{2}}}\right)}\  ��}-{{a_{3}}\ {b_{3}}\  ��}+{{a_{1}}\ {b_{1}}}\right)}\ {|_{\  1}}}+ 
\
\
\displaystyle
{{\frac{{\left({{\left({{a_{3}}\ {b_{4}}}-{{a_{4}}\ {b_{3}}}\right)}\  ��}+{{a_{1}}\ {b_{2}}}+{{a_{2}}\ {b_{1}}}\right)}\  ��}{\overline ��}}\ {|_{\  i}}}+ 
\
\
\displaystyle
{{\frac{{{\left(-{{a_{2}}\ {b_{4}}}+{{a_{4}}\ {b_{2}}}\right)}\  �� \  ��}+{{\left({{a_{1}}\ {b_{3}}}+{{a_{3}}\ {b_{1}}}\right)}\  ��}}{\overline ��}}\ {|_{\  j}}}+ 
\
\
\displaystyle
{{\left({{a_{1}}\ {b_{4}}}+{{a_{2}}\ {b_{3}}}-{{a_{3}}\ {b_{2}}}+{{a_{4}}\ {b_{1}}}\right)}\ {|_{\  k}}}
\end{array}
\end{equation*}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

Multiplication is Associative
fricas
test(
  ( I Y ) / _
  (  Y  ) = _
  ( Y I ) / _
  (  Y  ) )
\begin{equation}
\label{eq20} \mbox{\rm false} \end{equation}
Type: Boolean

A scalar product is denoted by the (2,0)-tensor
$U = \{ u_{ij} \}$
fricas
U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)
\begin{equation}
\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}}}
\end{array}
\end{equation}
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:
$$
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$$
  \begin{equation}
\label{eq22}
  \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.
fricas
ω:𝐋 :=
     (    Y I    )  /
           U        -
     (    I Y    )  /
           U
\begin{equation*}
\label{eq23}\begin{array}{@{}l}
\displaystyle
{{\frac{{{u^{1, \: 2}}\ {\overline ��}}-{{u^{1, \: 2}}\  ��}}{\overline ��}}\ {|^{\  1 \  1 \  i}}}+{{\frac{{{u^{1, \: 3}}\ {\overline ��}}-{{u^{1, \: 3}}\  ��}}{\overline ��}}\ {|^{\  1 \  1 \  j}}}+ 
\
\
\displaystyle
{{\frac{{\left({u^{2, \: 1}}-{u^{1, \: 2}}\right)}\  ��}{\overline ��}}\ {|^{\  1 \  i \  1}}}+{{\frac{{{u^{1, \: 1}}\  �� \ {\overline ��}}+{{u^{2, \: 2}}\  ��}}{\overline ��}}\ {|^{\  1 \  i \  i}}}+ 
\
\
\displaystyle
{{\frac{-{{u^{1, \: 4}}\ {\overline ��}}+{{u^{2, \: 3}}\  ��}}{\overline ��}}\ {|^{\  1 \  i \  j}}}+{{\frac{{{u^{1, \: 3}}\  �� \  �� \ {\overline ��}}+{{u^{2, \: 4}}\  �� \ {\overline ��}}}{{\overline ��}\ {\overline ��}}}\ {|^{\  1 \  i \  k}}}+ 
\
\
\displaystyle
{{\frac{{\left({u^{3, \: 1}}-{u^{1, \: 3}}\right)}\  ��}{\overline ��}}\ {|^{\  1 \  j \  1}}}+{{\frac{{{u^{1, \: 4}}\ {\overline ��}}+{{u^{3, \: 2}}\  ��}}{\overline ��}}\ {|^{\  1 \  j \  i}}}+ 
\
\
\displaystyle
{{\frac{{{u^{1, \: 1}}\  �� \ {\overline ��}}+{{u^{3, \: 3}}\  ��}}{\overline ��}}\ {|^{\  1 \  j \  j}}}+{{\frac{{{u^{3, \: 4}}\  �� \ {\overline ��}}-{{u^{1, \: 2}}\  �� \  �� \ {\overline ��}}}{{\overline ��}\ {\overline ��}}}\ {|^{\  1 \  j \  k}}}+ 
\
\
\displaystyle
{{\left({u^{4, \: 1}}-{u^{1, \: 4}}\right)}\ {|^{\  1 \  k \  1}}}+{{\frac{{{u^{4, \: 2}}\ {\overline ��}}-{{u^{1, \: 3}}\  �� \  ��}}{\overline ��}}\ {|^{\  1 \  k \  i}}}+ 
\
\
\displaystyle
{{\frac{{{u^{4, \: 3}}\ {\overline ��}}+{{u^{1, \: 2}}\  �� \  ��}}{\overline ��}}\ {|^{\  1 \  k \  j}}}+{{\left({{u^{1, \: 1}}\  �� \  ��}+{u^{4, \: 4}}\right)}\ {|^{\  1 \  k \  k}}}+ 
\
\
\displaystyle
{{\frac{-{{u^{2, \: 1}}\ {\overline ��}}+{{u^{2, \: 1}}\  ��}}{\overline ��}}\ {|^{\  i \  1 \  1}}}+ 
\
\
\displaystyle
{{\frac{-{{u^{2, \: 3}}\  �� \ {\overline ��}}+{{u^{2, \: 3}}\  �� \ {\overline ��}}}{{\overline ��}\ {\overline ��}}}\ {|^{\  i \  1 \  j}}}+ 
\
\
\displaystyle
{{\frac{-{{u^{2, \: 4}}\ {\overline ��}}+{{u^{2, \: 4}}\  ��}}{\overline ��}}\ {|^{\  i \  1 \  k}}}+{{\frac{-{{u^{1, \: 1}}\  �� \ {\overline ��}}-{{u^{2, \: 2}}\  ��}}{\overline ��}}\ {|^{\  i \  i \  1}}}+ 
\
\
\displaystyle
{{\left({u^{2, \: 1}}-{u^{1, \: 2}}\right)}\  �� \ {|^{\  i \  i \  i}}}+{{\left(-{{u^{1, \: 3}}\  ��}-{u^{2, \: 4}}\right)}\ {|^{\  i \  i \  j}}}+ 
\
\
\displaystyle
{{\frac{-{{u^{1, \: 4}}\  �� \ {\overline ��}}+{{u^{2, \: 3}}\  �� \  ��}}{\overline ��}}\ {|^{\  i \  i \  k}}}+{{\frac{{{u^{4, \: 1}}\ {\overline ��}}-{{u^{2, \: 3}}\  ��}}{\overline ��}}\ {|^{\  i \  j \  1}}}+ 
\
\
\displaystyle
{{\left({u^{4, \: 2}}+{u^{2, \: 4}}\right)}\ {|^{\  i \  j \  i}}}+{{\left({{u^{2, \: 1}}\  ��}+{u^{4, \: 3}}\right)}\ {|^{\  i \  j \  j}}}+ 
\
\
\displaystyle
{{\frac{{{u^{4, \: 4}}\ {\overline ��}}-{{u^{2, \: 2}}\  �� \  ��}}{\overline ��}}\ {|^{\  i \  j \  k}}}+{{\frac{-{{u^{2, \: 4}}\ {\overline ��}}-{{u^{3, \: 1}}\  �� \  ��}}{\overline ��}}\ {|^{\  i \  k \  1}}}+ 
\
\
\displaystyle
{{\frac{{\left(-{u^{3, \: 2}}-{u^{2, \: 3}}\right)}\  �� \  ��}{\overline ��}}\ {|^{\  i \  k \  i}}}+ 
\
\
\displaystyle
{{\frac{-{{u^{3, \: 3}}\  �� \  �� \ {\overline ��}}+{{u^{2, \: 2}}\  �� \  �� \ {\overline ��}}}{{\overline ��}\ {\overline ��}}}\ {|^{\  i \  k \  j}}}+ 
\
\
\displaystyle
{{\frac{{{u^{2, \: 1}}\  �� \  �� \ {\overline ��}}-{{u^{3, \: 4}}\  �� \  ��}}{\overline ��}}\ {|^{\  i \  k \  k}}}+{{\frac{-{{u^{3, \: 1}}\ {\overline ��}}+{{u^{3, \: 1}}\  ��}}{\overline ��}}\ {|^{\  j \  1 \  1}}}+ 
\
\
\displaystyle
{{\frac{{{u^{3, \: 2}}\  �� \ {\overline ��}}-{{u^{3, \: 2}}\  �� \ {\overline ��}}}{{\overline ��}\ {\overline ��}}}\ {|^{\  j \  1 \  i}}}+{{\frac{-{{u^{3, \: 4}}\ {\overline ��}}+{{u^{3, \: 4}}\  ��}}{\overline ��}}\ {|^{\  j \  1 \  k}}}+ 
\
\
\displaystyle
{{\frac{-{{u^{4, \: 1}}\ {\overline ��}}-{{u^{3, \: 2}}\  ��}}{\overline ��}}\ {|^{\  j \  i \  1}}}+{{\left({{u^{3, \: 1}}\  ��}-{u^{4, \: 2}}\right)}\ {|^{\  j \  i \  i}}}+ 
\
\
\displaystyle
{{\left(-{u^{4, \: 3}}-{u^{3, \: 4}}\right)}\ {|^{\  j \  i \  j}}}+{{\frac{-{{u^{4, \: 4}}\ {\overline ��}}+{{u^{3, \: 3}}\  �� \  ��}}{\overline ��}}\ {|^{\  j \  i \  k}}}+ 
\
\
\displaystyle
{{\frac{-{{u^{1, \: 1}}\  �� \ {\overline ��}}-{{u^{3, \: 3}}\  ��}}{\overline ��}}\ {|^{\  j \  j \  1}}}+{{\left(-{{u^{1, \: 2}}\  ��}+{u^{3, \: 4}}\right)}\ {|^{\  j \  j \  i}}}+ \
\
\displaystyle
{{\left({u^{3, \: 1}}-{u^{1, \: 3}}\right)}\  �� \ {|^{\  j \  j \  j}}}+{{\frac{-{{u^{1, \: 4}}\  �� \ {\overline ��}}-{{u^{3, \: 2}}\  �� \  ��}}{\overline ��}}\ {|^{\  j \  j \  k}}}+ 
\
\
\displaystyle
{{\frac{-{{u^{3, \: 4}}\ {\overline ��}}+{{u^{2, \: 1}}\  �� \  ��}}{\overline ��}}\ {|^{\  j \  k \  1}}}+ 
\
\
\displaystyle
{{\frac{-{{u^{3, \: 3}}\  �� \  �� \ {\overline ��}}+{{u^{2, \: 2}}\  �� \  �� \ {\overline ��}}}{{\overline ��}\ {\overline ��}}}\ {|^{\  j \  k \  i}}}+ 
\
\
\displaystyle
{{\frac{{\left({u^{3, \: 2}}+{u^{2, \: 3}}\right)}\  �� \  ��}{\overline ��}}\ {|^{\  j \  k \  j}}}+{{\frac{{{u^{3, \: 1}}\  �� \  �� \ {\overline ��}}+{{u^{2, \: 4}}\  �� \  ��}}{\overline ��}}\ {|^{\  j \  k \  k}}}+ 
\
\
\displaystyle
{{\frac{{{u^{4, \: 2}}\ {\overline ��}}-{{u^{4, \: 2}}\  ��}}{\overline ��}}\ {|^{\  k \  1 \  i}}}+{{\frac{{{u^{4, \: 3}}\ {\overline ��}}-{{u^{4, \: 3}}\  ��}}{\overline ��}}\ {|^{\  k \  1 \  j}}}+ 
\
\
\displaystyle
{{\frac{{{u^{3, \: 1}}\  �� \  �� \ {\overline ��}}-{{u^{4, \: 2}}\  �� \ {\overline ��}}}{{\overline ��}\ {\overline ��}}}\ {|^{\  k \  i \  1}}}+{{\frac{{{u^{4, \: 1}}\  �� \ {\overline ��}}+{{u^{3, \: 2}}\  �� \  ��}}{\overline ��}}\ {|^{\  k \  i \  i}}}+ 
\
\
\displaystyle
{{\frac{-{{u^{4, \: 4}}\ {\overline ��}}+{{u^{3, \: 3}}\  �� \  ��}}{\overline ��}}\ {|^{\  k \  i \  j}}}+{{\frac{{\left({u^{4, \: 3}}+{u^{3, \: 4}}\right)}\  �� \  ��}{\overline ��}}\ {|^{\  k \  i \  k}}}+ 
\
\
\displaystyle
{{\frac{-{{u^{4, \: 3}}\  �� \ {\overline ��}}-{{u^{2, \: 1}}\  �� \  �� \ {\overline ��}}}{{\overline ��}\ {\overline ��}}}\ {|^{\  k \  j \  1}}}+ 
\
\
\displaystyle
{{\frac{{{u^{4, \: 4}}\ {\overline ��}}-{{u^{2, \: 2}}\  �� \  ��}}{\overline ��}}\ {|^{\  k \  j \  i}}}+{{\frac{{{u^{4, \: 1}}\  �� \ {\overline ��}}-{{u^{2, \: 3}}\  �� \  ��}}{\overline ��}}\ {|^{\  k \  j \  j}}}+ 
\
\
\displaystyle
{{\frac{{\left(-{u^{4, \: 2}}-{u^{2, \: 4}}\right)}\  �� \  ��}{\overline ��}}\ {|^{\  k \  j \  k}}}+{{\left(-{{u^{1, \: 1}}\  �� \  ��}-{u^{4, \: 4}}\right)}\ {|^{\  k \  k \  1}}}+ 
\
\
\displaystyle
{{\frac{-{{u^{1, \: 2}}\  �� \  �� \ {\overline ��}}-{{u^{4, \: 3}}\  �� \  ��}}{\overline ��}}\ {|^{\  k \  k \  i}}}+ 
\
\
\displaystyle
{{\frac{-{{u^{1, \: 3}}\  �� \  �� \ {\overline ��}}+{{u^{4, \: 2}}\  �� \  ��}}{\overline ��}}\ {|^{\  k \  k \  j}}}+{{\left({u^{4, \: 1}}-{u^{1, \: 4}}\right)}\  �� \  �� \ {|^{\  k \  k \  k}}}
\end{array}
\end{equation*}
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
fricas
Ú:=
   (  Y Λ  ) / _
   (   Y I ) / _
        V
\begin{equation*}
\label{eq24}\begin{array}{@{}l}
\displaystyle
{{\frac{{{\left({2 \ {\overline ��}}+ �� \right)}\ {\overline ��}}+{�� \ {\overline ��}}}{{\overline ��}\ {\overline ��}}}\ {|^{\  1 \  1}}}+{{\frac{{{\left(-{2 \  �� \ {\overline ��}}-{�� \  ��}\right)}\ {\overline ��}}-{{{��}^{2}}\ {\overline ��}}}{{\overline ��}\ {\overline ��}}}\ {|^{\  i \  i}}}+ \
\
\displaystyle
{{\frac{{{\left(-{2 \  �� \ {\overline ��}}-{{��}^{2}}\right)}\ {\overline ��}}-{�� \  �� \ {\overline ��}}}{{\overline ��}\ {\overline ��}}}\ {|^{\  j \  j}}}+{{\frac{{{\left(-{2 \  �� \  �� \ {\overline ��}}-{{{��}^{2}}\  ��}\right)}\ {\overline ��}}-{�� \ {{��}^{2}}\ {\overline ��}}}{{\overline ��}\ {\overline ��}}}\ {|^{\  k \  k}}}
\end{array}
\end{equation*}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
Ù:=
   (  Λ Y  ) / _
   ( I Y   ) / _
      V
\begin{equation*}
\label{eq25}\begin{array}{@{}l}
\displaystyle
{{\frac{{{\left({2 \ {\overline ��}}+ �� \right)}\ {\overline ��}}+{�� \ {\overline ��}}}{{\overline ��}\ {\overline ��}}}\ {|^{\  1 \  1}}}+{{\frac{{{\left(-{2 \  �� \ {\overline ��}}-{�� \  ��}\right)}\ {\overline ��}}-{{{��}^{2}}\ {\overline ��}}}{{\overline ��}\ {\overline ��}}}\ {|^{\  i \  i}}}+ \
\
\displaystyle
{{\frac{{{\left(-{2 \  �� \ {\overline ��}}-{{��}^{2}}\right)}\ {\overline ��}}-{�� \  �� \ {\overline ��}}}{{\overline ��}\ {\overline ��}}}\ {|^{\  j \  j}}}+{{\frac{{{\left(-{2 \  �� \  �� \ {\overline ��}}-{{{��}^{2}}\  ��}\right)}\ {\overline ��}}-{�� \ {{��}^{2}}\ {\overline ��}}}{{\overline ��}\ {\overline ��}}}\ {|^{\  k \  k}}}
\end{array}
\end{equation*}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
test(Ù=Ú)
\begin{equation}
\label{eq26} \mbox{\rm true} \end{equation}
Type: Boolean

forms a non-degenerate associative scalar product for Y
fricas
Ũ := Ù
\begin{equation*}
\label{eq27}\begin{array}{@{}l}
\displaystyle
{{\frac{{{\left({2 \ {\overline ��}}+ �� \right)}\ {\overline ��}}+{�� \ {\overline ��}}}{{\overline ��}\ {\overline ��}}}\ {|^{\  1 \  1}}}+{{\frac{{{\left(-{2 \  �� \ {\overline ��}}-{�� \  ��}\right)}\ {\overline ��}}-{{{��}^{2}}\ {\overline ��}}}{{\overline ��}\ {\overline ��}}}\ {|^{\  i \  i}}}+ \
\
\displaystyle
{{\frac{{{\left(-{2 \  �� \ {\overline ��}}-{{��}^{2}}\right)}\ {\overline ��}}-{�� \  �� \ {\overline ��}}}{{\overline ��}\ {\overline ��}}}\ {|^{\  j \  j}}}+{{\frac{{{\left(-{2 \  �� \  �� \ {\overline ��}}-{{{��}^{2}}\  ��}\right)}\ {\overline ��}}-{�� \ {{��}^{2}}\ {\overline ��}}}{{\overline ��}\ {\overline ��}}}\ {|^{\  k \  k}}}
\end{array}
\end{equation*}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
test
     (    Y I    )  /
           Ũ        =
     (    I Y    )  /
           Ũ
\begin{equation}
\label{eq28} \mbox{\rm false} \end{equation}
Type: Boolean
fricas
determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ), j,1..dim), i,1..dim)
\begin{equation*}
\label{eq29}\frac{{{\left(-{{16}\ {{��}^{2}}\ {{��}^{2}}\ {{\overline ��}^{4}}}-{{32}\ {{��}^{3}}\ {{��}^{2}}\ {{\overline ��}^{3}}}-{{24}\ {{��}^{4}}\ {{��}^{2}}\ {{\overline ��}^{2}}}-{8 \ {{��}^{5}}\ {{��}^{2}}\ {\overline ��}}-{{{��}^{6}}\ {{��}^{2}}}\right)}\ {{\overline ��}^{4}}}+{{\left(-{{32}\ {{��}^{2}}\ {{��}^{3}}\ {{\overline ��}^{4}}}-{{48}\ {{��}^{3}}\ {{��}^{3}}\ {{\overline ��}^{3}}}-{{24}\ {{��}^{4}}\ {{��}^{3}}\ {{\overline ��}^{2}}}-{4 \ {{��}^{5}}\ {{��}^{3}}\ {\overline ��}}\right)}\ {{\overline ��}^{3}}}+{{\left(-{{24}\ {{��}^{2}}\ {{��}^{4}}\ {{\overline ��}^{4}}}-{{24}\ {{��}^{3}}\ {{��}^{4}}\ {{\overline ��}^{3}}}-{6 \ {{��}^{4}}\ {{��}^{4}}\ {{\overline ��}^{2}}}\right)}\ {{\overline ��}^{2}}}+{{\left(-{8 \ {{��}^{2}}\ {{��}^{5}}\ {{\overline ��}^{4}}}-{4 \ {{��}^{3}}\ {{��}^{5}}\ {{\overline ��}^{3}}}\right)}\ {\overline ��}}-{{{��}^{2}}\ {{��}^{6}}\ {{\overline ��}^{4}}}}{{{\overline ��}^{4}}\ {{\overline ��}^{4}}}\end{equation*}
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.
fricas
)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))
fricas
u := transpose matrix [concat map(variables,ravel U)::ℒ Symbol];
Type: Matrix(Polynomial(Integer))
fricas
J::OutputForm * u::OutputForm = 0
\begin{equation*}
\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 &{\frac{{\overline ��}- ��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 &{\frac{{\overline ��}- ��}{\overline ��}}& 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 & -{\frac{��}{\overline ��}}& 0 & 0 &{\frac{��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
�� & 0 & 0 & 0 & 0 &{\frac{��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & - 1 & 0 & 0 &{\frac{��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 &{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 &{\frac{��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & -{\frac{��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 &{\frac{��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 &{\frac{��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & 0 
\
�� & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 
\
0 & -{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{��}{\overline ��}}& 0 & 0 & 0 & 0 
\
0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 
\
0 & 0 & -{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 
\
0 &{\frac{�� \  ��}{\overline ��}}& 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 & 1 
\
0 & 0 & 0 & 0 &{\frac{-{\overline ��}+ ��}{\overline ��}}& 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 &{\frac{-{�� \ {\overline ��}}+{�� \ {\overline ��}}}{{\overline ��}\ {\overline ��}}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{-{\overline ��}+ ��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
- �� & 0 & 0 & 0 & 0 & -{\frac{��}{\overline ��}}& 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 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & - �� & 0 & 0 &{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & -{\frac{��}{\overline ��}}& 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 & �� & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 
\
0 & 0 & 0 & 0 & 0 & -{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & -{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & -{\frac{�� \  ��}{\overline ��}}& 0 & 0 & -{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 &{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 & -{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 &{�� \  ��}& 0 & 0 & 0 & 0 & 0 & 0 & -{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{-{\overline ��}+ ��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{{�� \ {\overline ��}}-{�� \ {\overline ��}}}{{\overline ��}\ {\overline ��}}}& 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 &{\frac{-{\overline ��}+ ��}{\overline ��}}& 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -{\frac{��}{\overline ��}}& 0 & 0 & - 1 & 0 & 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 & - 1 & 0 & 0 & - 1 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 & - 1 
\
- �� & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -{\frac{��}{\overline ��}}& 0 & 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 & -{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 &{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 &{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 & -{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 &{\frac{�� \  ��}{\overline ��}}& 0 & 0 &{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{�� \  ��}{\overline ��}}&{�� \  ��}& 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 &{\frac{{\overline ��}- ��}{\overline ��}}& 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{{\overline ��}- ��}{\overline ��}}& 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 &{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 & -{\frac{��}{\overline ��}}& 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{�� \  ��}{\overline ��}}& 0 & 0 & �� & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 & - 1 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{�� \  ��}{\overline ��}}& 0 & 0 &{\frac{�� \  ��}{\overline ��}}& 0 
\
0 & 0 & 0 & 0 & -{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -{\frac{��}{\overline ��}}& 0 
\
0 & 0 & 0 & 0 & 0 & -{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 
\
0 & 0 & 0 & 0 & 0 & 0 & -{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & �� & 0 & 0 & 0 
\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -{\frac{�� \  ��}{\overline ��}}& 0 & 0 & 0 & 0 & 0 & -{\frac{�� \  ��}{\overline ��}}& 0 & 0 
\
-{�� \  ��}& 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 & -{\frac{�� \  ��}{\overline ��}}& 0 
\
0 & 0 & -{�� \  ��}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{\frac{�� \  ��}{\overline ��}}& 0 & 0 
\
0 & 0 & 0 & -{�� \  ��}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &{�� \  ��}& 0 & 0 & 0 
\end{array}
\right]}\ {\left[ 
\begin{array}{c}
{u^{1, \: 1}}
\
{u^{1, \: 2}}
\
{u^{1, \: 3}}
\
{u^{1, \: 4}}
\
{u^{2, \: 1}}
\
{u^{2, \: 2}}
\
{u^{2, \: 3}}
\
{u^{2, \: 4}}
\
{u^{3, \: 1}}
\
{u^{3, \: 2}}
\
{u^{3, \: 3}}
\
{u^{3, \: 4}}
\
{u^{4, \: 1}}
\
{u^{4, \: 2}}
\
{u^{4, \: 3}}
\
{u^{4, \: 4}}
\end{array}
\right]}}= 
\
\
\displaystyle
0 
\end{array}
\end{equation*}
Type: Equation(OutputForm?)
fricas
nrows(J),ncols(J)
\begin{equation*}
\label{eq31}\left[{64}, \:{16}\right]?\end{equation*}
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:
fricas
Ñ:=nullSpace(J)
\begin{equation*}
\label{eq32}\left[{\left[ -{\frac{1}{�� \  ��}}, \: 0, \: 0, \: 0, \: 0, \:{\frac{\overline ��}{�� \  ��}}, \: 0, \: 0, \: 0, \: 0, \:{\frac{\overline ��}{�� \  ��}}, \: 0, \: 0, \: 0, \: 0, \: 1 \right]?}\right]\end{equation*}
Type: List(Vector(Expression(Integer)))
fricas
ℰ:=map((x,y)+->x=y, concat
       map(variables,ravel U), entries Σ(sb('p,[i])*Ñ.i, i,1..#Ñ) )
\begin{equation*}
\label{eq33}\begin{array}{@{}l}
\displaystyle
\left[{{u^{1, \: 1}}= -{\frac{p_{1}}{�� \  ��}}}, \:{{u^{1, \: 2}}= 0}, \:{{u^{1, \: 3}}= 0}, \:{{u^{1, \: 4}}= 0}, \:{{u^{2, \: 1}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{2, \: 2}}={\frac{{p_{1}}\ {\overline ��}}{�� \  ��}}}, \:{{u^{2, \: 3}}= 0}, \:{{u^{2, \: 4}}= 0}, \:{{u^{3, \: 1}}= 0}, \:{{u^{3, \: 2}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{3, \: 3}}={\frac{{p_{1}}\ {\overline ��}}{�� \  ��}}}, \:{{u^{3, \: 4}}= 0}, \:{{u^{4, \: 1}}= 0}, \:{{u^{4, \: 2}}= 0}, \:{{u^{4, \: 3}}= 0}, \: \right.
\
\
\displaystyle
\left.{{u^{4, \: 4}}={p_{1}}}\right] 
\end{array}
\end{equation*}
Type: List(Equation(Expression(Integer)))

This defines a family of pre-Frobenius algebras:
fricas
zero? eval(ω,ℰ)
\begin{equation}
\label{eq34} \mbox{\rm true} \end{equation}
Type: Boolean

In general the pairing is not symmetric!
fricas
Ų:𝐋 := eval(U,ℰ)
\begin{equation*}
\label{eq35}-{{\frac{p_{1}}{�� \  ��}}\ {|^{\  1 \  1}}}+{{\frac{{p_{1}}\ {\overline ��}}{�� \  ��}}\ {|^{\  i \  i}}}+{{\frac{{p_{1}}\ {\overline ��}}{�� \  ��}}\ {|^{\  j \  j}}}+{{p_{1}}\ {|^{\  k \  k}}}\end{equation*}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
matrix Ξ(Ξ((𝐞.i 𝐞.j)/Ų, i,1..dim), j,1..dim)
\begin{equation*}
\label{eq36}\left[ 
\begin{array}{cccc}
-{\frac{p_{1}}{�� \  ��}}& 0 & 0 & 0 
\
0 &{\frac{{p_{1}}\ {\overline ��}}{�� \  ��}}& 0 & 0 
\
0 & 0 &{\frac{{p_{1}}\ {\overline ��}}{�� \  ��}}& 0 
\
0 & 0 & 0 &{p_{1}}
\end{array}
\right]\end{equation*}
Type: Matrix(ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer)))

This is the most general form of the "dot product" of two quaternions
fricas
(a*b)/Ų
\begin{equation*}
\label{eq37}\frac{{{a_{2}}\ {b_{2}}\ {p_{1}}\ {\overline ��}}+{{a_{3}}\ {b_{3}}\ {p_{1}}\ {\overline ��}}+{{a_{4}}\ {b_{4}}\ {p_{1}}\  �� \  ��}-{{a_{1}}\ {b_{1}}\ {p_{1}}}}{�� \  ��}\end{equation*}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
(a*a)/Ų
\begin{equation*}
\label{eq38}\frac{{{{a_{2}}^{2}}\ {p_{1}}\ {\overline ��}}+{{{a_{3}}^{2}}\ {p_{1}}\ {\overline ��}}+{{{a_{4}}^{2}}\ {p_{1}}\  �� \  ��}-{{{a_{1}}^{2}}\ {p_{1}}}}{�� \  ��}\end{equation*}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

Cartan-Killing is a special case
fricas
ck:=solve(equate(Ũ=Ų),Ξ(sb('p,[i]), i,1..#Ñ)).1
fricas
Compiling function equate with type Equation(ClosedLinearOperator(
      OrderedVariableList([1,i,j,k]),Expression(Integer))) -> List(
      Equation(Expression(Integer))) 

   >> Error detected within library code:
   index out of range

The scalar product must be non-degenerate:
fricas
Ů:=determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ų), j,1..dim), i,1..dim)
\begin{equation*}
\label{eq39}-{\frac{{{p_{1}}^{4}}\ {\overline ��}\ {\overline ��}}{{{��}^{3}}\ {{��}^{3}}}}\end{equation*}
Type: Expression(Integer)
fricas
factor(numer Ů)/factor(denom Ů)
\begin{equation*}
\label{eq40}-{\frac{{{p_{1}}^{4}}\ {\overline ��}\ {\overline ��}}{{{��}^{3}}\ {{��}^{3}}}}\end{equation*}
Type: Fraction(Factored(SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer)))))

Definition 3
  Co-pairing
Solve the [Snake Relation]? as a system of linear equations.
fricas
Ω:𝐋:=Σ(Σ(script('u,[[i,j]])*𝐞.i*𝐞.j, i,1..dim), j,1..dim)
\begin{equation}
\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}}}
\end{array}
\end{equation}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
d1:=(I*Ω)/(Ų*I);
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
d2:=(Ω*I)/(I*Ų);
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
eq1:=equate(d1=I);
Type: List(Equation(Expression(Integer)))
fricas
eq2:=equate(d2=I);
Type: List(Equation(Expression(Integer)))
fricas
snake:=solve(concat(eq1,eq2),concat Ξ(Ξ(script('u,[[i,j]]), i,1..dim), j,1..dim));
Type: List(List(Equation(Expression(Integer))))
fricas
if #snake ~= 1 then error "no solution"
Type: Void
fricas
Ω:=eval(Ω,snake(1))
\begin{equation*}
\label{eq42}-{{\frac{�� \  ��}{p_{1}}}\ {|_{\  1 \  1}}}+{{\frac{�� \  ��}{{p_{1}}\ {\overline ��}}}\ {|_{\  i \  i}}}+{{\frac{�� \  ��}{{p_{1}}\ {\overline ��}}}\ {|_{\  j \  j}}}+{{\frac{1}{p_{1}}}\ {|_{\  k \  k}}}\end{equation*}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
matrix Ξ(Ξ(Ω/(𝐝.i*𝐝.j), i,1..dim), j,1..dim)
\begin{equation*}
\label{eq43}\left[ 
\begin{array}{cccc}
-{\frac{�� \  ��}{p_{1}}}& 0 & 0 & 0 
\
0 &{\frac{�� \  ��}{{p_{1}}\ {\overline ��}}}& 0 & 0 
\
0 & 0 &{\frac{�� \  ��}{{p_{1}}\ {\overline ��}}}& 0 
\
0 & 0 & 0 &{\frac{1}{p_{1}}}
\end{array}
\right]\end{equation*}
Type: Matrix(ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer)))

Check "dimension" and the snake relations.
fricas
d:𝐋:=
       Ω    /
       X    /
       Ų
\begin{equation}
\label{eq44}4\end{equation}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
test
    (    I Ω     )  /
    (     Ų I    )  =  I
\begin{equation}
\label{eq45} \mbox{\rm true} \end{equation}
Type: Boolean
fricas
test
    (     Ω I    )  /
    (    I Ų     )  =  I
\begin{equation}
\label{eq46} \mbox{\rm true} \end{equation}
Type: Boolean

Definition 4
  Co-algebra
Compute the "three-point" function and use it to define co-multiplication.
fricas
W:=(Y,I)/Ų
\begin{equation*}
\label{eq47}\begin{array}{@{}l}
\displaystyle
-{{\frac{p_{1}}{�� \  ��}}\ {|^{\  1 \  1 \  1}}}+{{\frac{p_{1}}{��}}\ {|^{\  1 \  i \  i}}}+{{\frac{p_{1}}{��}}\ {|^{\  1 \  j \  j}}}+ 
\
\
\displaystyle
{{p_{1}}\ {|^{\  1 \  k \  k}}}+{{\frac{p_{1}}{��}}\ {|^{\  i \  1 \  i}}}+{{\frac{p_{1}}{��}}\ {|^{\  i \  i \  1}}}+{{p_{1}}\ {|^{\  i \  j \  k}}}- 
\
\
\displaystyle
{{p_{1}}\ {|^{\  i \  k \  j}}}+{{\frac{p_{1}}{��}}\ {|^{\  j \  1 \  j}}}-{{p_{1}}\ {|^{\  j \  i \  k}}}+{{\frac{p_{1}}{��}}\ {|^{\  j \  j \  1}}}+ 
\
\
\displaystyle
{{p_{1}}\ {|^{\  j \  k \  i}}}+{{p_{1}}\ {|^{\  k \  1 \  k}}}+{{p_{1}}\ {|^{\  k \  i \  j}}}-{{p_{1}}\ {|^{\  k \  j \  i}}}+{{p_{1}}\ {|^{\  k \  k \  1}}}
\end{array}
\end{equation*}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
λ:=(Ω,I,Ω)/(I,W,I)
\begin{equation*}
\label{eq48}\begin{array}{@{}l}
\displaystyle
-{{\frac{�� \  ��}{p_{1}}}\ {|_{\  1 \  1}^{\  1}}}+{{\frac{�� \ {{��}^{2}}}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  i \  i}^{\  1}}}+{{\frac{{{��}^{2}}\  ��}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  j \  j}^{\  1}}}+ 
\
\
\displaystyle
{{\frac{1}{p_{1}}}\ {|_{\  k \  k}^{\  1}}}-{{\frac{�� \ {{��}^{2}}}{{p_{1}}\ {\overline ��}}}\ {|_{\  1 \  i}^{\  i}}}-{{\frac{�� \ {{��}^{2}}}{{p_{1}}\ {\overline ��}}}\ {|_{\  i \  1}^{\  i}}}- 
\
\
\displaystyle
{{\frac{�� \  ��}{{p_{1}}\ {\overline ��}}}\ {|_{\  j \  k}^{\  i}}}+{{\frac{�� \  ��}{{p_{1}}\ {\overline ��}}}\ {|_{\  k \  j}^{\  i}}}-{{\frac{{{��}^{2}}\  ��}{{p_{1}}\ {\overline ��}}}\ {|_{\  1 \  j}^{\  j}}}+ 
\
\
\displaystyle
{{\frac{�� \  ��}{{p_{1}}\ {\overline ��}}}\ {|_{\  i \  k}^{\  j}}}-{{\frac{{{��}^{2}}\  ��}{{p_{1}}\ {\overline ��}}}\ {|_{\  j \  1}^{\  j}}}-{{\frac{�� \  ��}{{p_{1}}\ {\overline ��}}}\ {|_{\  k \  i}^{\  j}}}- 
\
\
\displaystyle
{{\frac{�� \  ��}{p_{1}}}\ {|_{\  1 \  k}^{\  k}}}-{{\frac{{{��}^{2}}\ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  i \  j}^{\  k}}}+{{\frac{{{��}^{2}}\ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  j \  i}^{\  k}}}- 
\
\
\displaystyle
{{\frac{�� \  ��}{p_{1}}}\ {|_{\  k \  1}^{\  k}}}
\end{array}
\end{equation*}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

fricas
test
     (    I Ω     )  /
     (     Y I    )  =  λ
\begin{equation}
\label{eq49} \mbox{\rm true} \end{equation}
Type: Boolean
fricas
test
     (     Ω I    )  /
     (    I Y     )  =  λ
\begin{equation}
\label{eq50} \mbox{\rm true} \end{equation}
Type: Boolean

Co-associativity
fricas
test(
  (  λ  ) / _
  ( I λ ) = _
  (  λ  ) / _
  ( λ I ) )
\begin{equation}
\label{eq51} \mbox{\rm false} \end{equation}
Type: Boolean

Frobenius Condition (fork)
fricas
H :=
         Y    /
         λ
\begin{equation*}
\label{eq52}\begin{array}{@{}l}
\displaystyle
-{{\frac{�� \  ��}{p_{1}}}\ {|_{\  1 \  1}^{\  1 \  1}}}+{{\frac{�� \ {{��}^{2}}}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  i \  i}^{\  1 \  1}}}+ 
\
\
\displaystyle
{{\frac{{{��}^{2}}\  ��}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  j \  j}^{\  1 \  1}}}+{{\frac{1}{p_{1}}}\ {|_{\  k \  k}^{\  1 \  1}}}- 
\
\
\displaystyle
{{\frac{�� \ {{��}^{3}}}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  1 \  i}^{\  1 \  i}}}-{{\frac{�� \ {{��}^{3}}}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  i \  1}^{\  1 \  i}}}- 
\
\
\displaystyle
{{\frac{�� \ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  j \  k}^{\  1 \  i}}}+{{\frac{�� \ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  k \  j}^{\  1 \  i}}}- 
\
\
\displaystyle
{{\frac{{{��}^{3}}\  ��}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  1 \  j}^{\  1 \  j}}}+{{\frac{{{��}^{2}}\  ��}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  i \  k}^{\  1 \  j}}}- 
\
\
\displaystyle
{{\frac{{{��}^{3}}\  ��}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  j \  1}^{\  1 \  j}}}-{{\frac{{{��}^{2}}\  ��}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  k \  i}^{\  1 \  j}}}- 
\
\
\displaystyle
{{\frac{�� \  ��}{p_{1}}}\ {|_{\  1 \  k}^{\  1 \  k}}}-{{\frac{{{��}^{2}}\ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  i \  j}^{\  1 \  k}}}+ 
\
\
\displaystyle
{{\frac{{{��}^{2}}\ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  j \  i}^{\  1 \  k}}}-{{\frac{�� \  ��}{p_{1}}}\ {|_{\  k \  1}^{\  1 \  k}}}- 
\
\
\displaystyle
{{\frac{�� \ {{��}^{3}}}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  1 \  i}^{\  i \  1}}}-{{\frac{�� \ {{��}^{3}}}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  i \  1}^{\  i \  1}}}- 
\
\
\displaystyle
{{\frac{�� \ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  j \  k}^{\  i \  1}}}+{{\frac{�� \ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  k \  j}^{\  i \  1}}}+ 
\
\
\displaystyle
{{\frac{�� \ {{��}^{2}}}{p_{1}}}\ {|_{\  1 \  1}^{\  i \  i}}}-{{\frac{�� \ {{��}^{3}}}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  i \  i}^{\  i \  i}}}- 
\
\
\displaystyle
{{\frac{{{��}^{2}}\ {{��}^{2}}}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  j \  j}^{\  i \  i}}}-{{\frac{��}{p_{1}}}\ {|_{\  k \  k}^{\  i \  i}}}-{{\frac{�� \  ��}{p_{1}}}\ {|_{\  1 \  k}^{\  i \  j}}}- 
\
\
\displaystyle
{{\frac{{{��}^{2}}\ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  i \  j}^{\  i \  j}}}+{{\frac{{{��}^{2}}\ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  j \  i}^{\  i \  j}}}- 
\
\
\displaystyle
{{\frac{�� \  ��}{p_{1}}}\ {|_{\  k \  1}^{\  i \  j}}}+{{\frac{{{��}^{3}}\ {{��}^{2}}}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  1 \  j}^{\  i \  k}}}- 
\
\
\displaystyle
{{\frac{{{��}^{2}}\ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  i \  k}^{\  i \  k}}}+{{\frac{{{��}^{3}}\ {{��}^{2}}}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  j \  1}^{\  i \  k}}}+ 
\
\
\displaystyle
{{\frac{{{��}^{2}}\ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  k \  i}^{\  i \  k}}}-{{\frac{{{��}^{3}}\  ��}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  1 \  j}^{\  j \  1}}}+ 
\
\
\displaystyle
{{\frac{{{��}^{2}}\  ��}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  i \  k}^{\  j \  1}}}-{{\frac{{{��}^{3}}\  ��}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  j \  1}^{\  j \  1}}}- 
\
\
\displaystyle
{{\frac{{{��}^{2}}\  ��}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  k \  i}^{\  j \  1}}}+{{\frac{�� \  ��}{p_{1}}}\ {|_{\  1 \  k}^{\  j \  i}}}+ 
\
\
\displaystyle
{{\frac{{{��}^{2}}\ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  i \  j}^{\  j \  i}}}-{{\frac{{{��}^{2}}\ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  j \  i}^{\  j \  i}}}+ 
\
\
\displaystyle
{{\frac{�� \  ��}{p_{1}}}\ {|_{\  k \  1}^{\  j \  i}}}+{{\frac{{{��}^{2}}\  ��}{p_{1}}}\ {|_{\  1 \  1}^{\  j \  j}}}-{{\frac{{{��}^{2}}\ {{��}^{2}}}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  i \  i}^{\  j \  j}}}- 
\
\
\displaystyle
{{\frac{{{��}^{3}}\  ��}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  j \  j}^{\  j \  j}}}-{{\frac{��}{p_{1}}}\ {|_{\  k \  k}^{\  j \  j}}}- 
\
\
\displaystyle
{{\frac{{{��}^{2}}\ {{��}^{3}}}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  1 \  i}^{\  j \  k}}}-{{\frac{{{��}^{2}}\ {{��}^{3}}}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  i \  1}^{\  j \  k}}}- \
\
\displaystyle
{{\frac{{{��}^{2}}\ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  j \  k}^{\  j \  k}}}+{{\frac{{{��}^{2}}\ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  k \  j}^{\  j \  k}}}- 
\
\
\displaystyle
{{\frac{�� \  ��}{p_{1}}}\ {|_{\  1 \  k}^{\  k \  1}}}-{{\frac{{{��}^{2}}\ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  i \  j}^{\  k \  1}}}+ 
\
\
\displaystyle
{{\frac{{{��}^{2}}\ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  j \  i}^{\  k \  1}}}-{{\frac{�� \  ��}{p_{1}}}\ {|_{\  k \  1}^{\  k \  1}}}- 
\
\
\displaystyle
{{\frac{{{��}^{3}}\ {{��}^{2}}}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  1 \  j}^{\  k \  i}}}+{{\frac{{{��}^{2}}\ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  i \  k}^{\  k \  i}}}- 
\
\
\displaystyle
{{\frac{{{��}^{3}}\ {{��}^{2}}}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  j \  1}^{\  k \  i}}}-{{\frac{{{��}^{2}}\ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  k \  i}^{\  k \  i}}}+ 
\
\
\displaystyle
{{\frac{{{��}^{2}}\ {{��}^{3}}}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  1 \  i}^{\  k \  j}}}+{{\frac{{{��}^{2}}\ {{��}^{3}}}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  i \  1}^{\  k \  j}}}+ \
\
\displaystyle
{{\frac{{{��}^{2}}\ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  j \  k}^{\  k \  j}}}-{{\frac{{{��}^{2}}\ {{��}^{2}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  k \  j}^{\  k \  j}}}+ 
\
\
\displaystyle
{{\frac{{{��}^{2}}\ {{��}^{2}}}{p_{1}}}\ {|_{\  1 \  1}^{\  k \  k}}}-{{\frac{{{��}^{2}}\ {{��}^{3}}}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  i \  i}^{\  k \  k}}}- 
\
\
\displaystyle
{{\frac{{{��}^{3}}\ {{��}^{2}}}{{p_{1}}\ {{\overline ��}^{2}}}}\ {|_{\  j \  j}^{\  k \  k}}}-{{\frac{�� \  ��}{p_{1}}}\ {|_{\  k \  k}^{\  k \  k}}}
\end{array}
\end{equation*}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
test
     (   λ I   )  /
     (  I Y    )  =  H
\begin{equation}
\label{eq53} \mbox{\rm false} \end{equation}
Type: Boolean
fricas
test
     (   I λ   )  /
     (    Y I  )  =  H
\begin{equation}
\label{eq54} \mbox{\rm false} \end{equation}
Type: Boolean

The Cartan-Killing form makes H of the Frobenius condition idempotent
fricas
test( eval(H,ck)=eval(H/H,ck) )

   There are 10 exposed and 6 unexposed library operations named eval 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op eval
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named eval 
      with argument type(s) 
  ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
                                Variable(ck)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

But it is not unique. E.g. other idempots
fricas
h1:=map(numer,ravel(H-H/H)::List FRAC POLY INT);

   Cannot convert the value from type List(Expression(Integer)) to List
      (Fraction(Polynomial(Integer))) .

Handle
fricas
Φ:𝐋 :=
         λ     /
         Y
\begin{equation*}
\label{eq55}\begin{array}{@{}l}
\displaystyle
{{\frac{{{\left(-{2 \  �� \  �� \ {{\overline ��}^{2}}}-{{{��}^{3}}\  ��}\right)}\ {{\overline ��}^{2}}}-{�� \ {{��}^{3}}\ {{\overline ��}^{2}}}}{{p_{1}}\ {{\overline ��}^{2}}\ {{\overline ��}^{2}}}}\ {|_{\  1}^{\  1}}}+ 
\
\
\displaystyle
{{\frac{-{2 \ {{��}^{2}}\ {{��}^{2}}\ {\overline ��}}-{2 \  �� \ {{��}^{3}}\ {\overline ��}}}{{p_{1}}\ {\overline ��}\ {{\overline ��}^{2}}}}\ {|_{\  i}^{\  i}}}+ 
\
\
\displaystyle
{{\frac{-{2 \ {{��}^{3}}\  �� \ {\overline ��}}-{2 \ {{��}^{2}}\ {{��}^{2}}\ {\overline ��}}}{{p_{1}}\ {{\overline ��}^{2}}\ {\overline ��}}}\ {|_{\  j}^{\  j}}}+ 
\
\
\displaystyle
{{\frac{-{2 \  �� \  �� \ {\overline ��}\ {\overline ��}}-{2 \ {{��}^{2}}\ {{��}^{2}}}}{{p_{1}}\ {\overline ��}\ {\overline ��}}}\ {|_{\  k}^{\  k}}}
\end{array}
\end{equation*}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

The Cartan-Killing form makes Φ of the identity
fricas
test( eval(Φ,ck)=I )

   There are 10 exposed and 6 unexposed library operations named eval 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op eval
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named eval 
      with argument type(s) 
  ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
                                Variable(ck)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

but it can be the identity in many ways. For example,
fricas
solve(equate(eval(Φ,[p[1]=1,p[2]=1,p[3]=1,σ=1,μ=1])=I),[p[4]])
\begin{equation*}
\label{eq56}\left[ \right]?\end{equation*}
Type: List(List(Equation(Expression(Integer))))

If handle is identity then fork is idempotent but the converse is not true
fricas
Φ1:=map(numer,ravel(Φ-I)::List FRAC POLY INT);

   Cannot convert the value from type List(Expression(Integer)) to List
      (Fraction(Polynomial(Integer))) .

Bi-algebra conditions
fricas
ΦΦ:=         _
  (  λ λ  ) / _
  ( I X I ) / _
  (  Y Y  ) ;
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
bi1:=map(numer,ravel(ΦΦ-H)::List FRAC POLY INT);

   Cannot convert the value from type List(Expression(Integer)) to List
      (Fraction(Polynomial(Integer))) .

Definition 5
  i = Unit of the algebra
fricas
i:=𝐞.1
\begin{equation}
\label{eq57}|_{\  1}\end{equation}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
test
         i     /
         λ     =    Ω
\begin{equation}
\label{eq58} \mbox{\rm false} \end{equation}
Type: Boolean

Co-unit
i 
U

fricas
ι:𝐋:=
    (    i I   ) /
          Ų
\begin{equation*}
\label{eq59}-{{\frac{p_{1}}{�� \  ��}}\ {|^{\  1}}}\end{equation*}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

Y=U
ι  

fricas
test
        Y     /
        ι     =  Ų
\begin{equation}
\label{eq60} \mbox{\rm false} \end{equation}
Type: Boolean

For example:
fricas
ex1:=[σ=1,μ=1,p[1]=1,p[2]=1,p[3]=1,p[4]=1]
\begin{equation*}
\label{eq61}\left[{�� = 1}, \:{�� = 1}, \:{{p_{1}}= 1}, \:{{p_{2}}= 1}, \:{{p_{3}}= 1}, \:{{p_{4}}= 1}\right]?\end{equation*}
Type: List(Equation(Polynomial(Integer)))
fricas
Ų0:𝐋  :=eval(Ų,ex1)
\begin{equation}
\label{eq62}-{|^{\  1 \  1}}+{|^{\  i \  i}}+{|^{\  j \  j}}+{|^{\  k \  k}}\end{equation}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
Ω0:𝐋  :=eval(Ω,ex1)$𝐋
\begin{equation}
\label{eq63}-{|_{\  1 \  1}}+{|_{\  i \  i}}+{|_{\  j \  j}}+{|_{\  k \  k}}\end{equation}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
λ0:𝐋  :=eval(λ,ex1)$𝐋
\begin{equation}
\label{eq64}\begin{array}{@{}l}
\displaystyle
-{|_{\  1 \  1}^{\  1}}+{|_{\  i \  i}^{\  1}}+{|_{\  j \  j}^{\  1}}+{|_{\  k \  k}^{\  1}}-{|_{\  1 \  i}^{\  i}}-{|_{\  i \  1}^{\  i}}- 
\
\
\displaystyle
{|_{\  j \  k}^{\  i}}+{|_{\  k \  j}^{\  i}}-{|_{\  1 \  j}^{\  j}}+{|_{\  i \  k}^{\  j}}-{|_{\  j \  1}^{\  j}}-{|_{\  k \  i}^{\  j}}-{|_{\  1 \  k}^{\  k}}- 
\
\
\displaystyle
{|_{\  i \  j}^{\  k}}+{|_{\  j \  i}^{\  k}}-{|_{\  k \  1}^{\  k}}
\end{array}
\end{equation}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))
fricas
Φ0:𝐋 :=eval(Φ,ex1)$𝐋
\begin{equation}
\label{eq65}-{4 \ {|_{\  1}^{\  1}}}-{4 \ {|_{\  i}^{\  i}}}-{4 \ {|_{\  j}^{\  j}}}-{4 \ {|_{\  k}^{\  k}}}\end{equation}
Type: ClosedLinearOperator?(OrderedVariableList?([1,i,j,k]),Expression(Integer))

! Argument of �has an extra }.
<inserted text> 
                \par 
l.179 ...t[{\left[ 0, \:{\frac{��}{\overline ��}
                                                  }, \: 0, \: 0 \right]}, \:...

Runaway argument?
 Paragraph ended before �was complete.
<to be read again> 
                   \par 
l.179 ...t[{\left[ 0, \:{\frac{��}{\overline ��}
                                                  }, \: 0, \: 0 \right]}, \:...
 Package utf8x Error: Character131appearedalone.

See the utf8x package documentation for explanation.
Type  H <return>  for immediate help.
 ...                                              
l.179 ...t[{\left[ 0, \:{\frac{��}{\overline ��}
                                                  }, \: 0, \: 0 \right]}, \:...
 Argument of �has an extra }.
<inserted text> 
                \par 
l.179 ...ht]}, \:{\left[{\frac{��}{\overline ��}
                                                  }, \: 0, \: 0, \: 0 \right...

Runaway argument?
 Paragraph ended before �was complete.
<to be read again> 
                   \par 
l.179 ...ht]}, \:{\left[{\frac{��}{\overline ��}
                                                  }, \: 0, \: 0, \: 0 \right...
 Package utf8x Error: Character131appearedalone.

See the utf8x package documentation for explanation.
Type  H <return>  for immediate help.
 ...                                              
l.179 ...ht]}, \:{\left[{\frac{��}{\overline ��}
                                                  }, \: 0, \: 0, \: 0 \right...
 Argument of �has an extra }.
<inserted text> 
                \par 
l.179 ... 0, \: -{\frac{�� \  ��}{\overline ��}
                                                  }\right]}, \:{\left[ 0, \:...

Runaway argument?
 Paragraph ended before �was complete.
<to be read again> 
                   \par 
l.179 ... 0, \: -{\frac{�� \  ��}{\overline ��}
                                                  }\right]}, \:{\left[ 0, \:...
 Package utf8x Error: Character131appearedalone.

See the utf8x package documentation for explanation.
Type  H <return>  for immediate help.
 ...                                              
l.179 ... 0, \: -{\frac{�� \  ��}{\overline ��}
                                                  }\right]}, \:{\left[ 0, \:...
 Argument of �has an extra }.
<inserted text> 
                \par 
l.179 ...\: 0, \:{\frac{�� \  ��}{\overline ��}
                                                  }, \: 0 \right]}\right]
Runaway argument?