\documentclass{article}$

\usepackage{amsmath,amssymb}$

\sloppy$

\begin{document}$

This output from the file \texttt{program13_45case2_2.red}.\\$

Computation of all complex    structures$

up to equivalence on the real Lie Algebra$

USD {\mathcal{G}}_{6,m5}.USD$

in the case 2 where B= C = 0 and in the subcase 2_2 where xi(5,5) neq 0.$

\smallskip  \par $

Commutation relations for$

USD {\mathcal{G}}_{6,m5}:USD\\$

USD[x(1),x(3)]=x(5)USD;$

USD[x(1),x(4)]=x(6)USD;$

USD[x(2),x(3)]= - x(6)USD;$

USD[x(2),x(4)]=x(5)USD;$

\P$

Nonzero torsion$

\par$

Torsion equations to cancel (Latex output) : \\USD$

{1,2}|1\\xi(3,2)*xi(1,5) + xi(3,1)*xi(1,6) - xi(4,1)*xi(1,5) + xi(4,2)*xi(1,6)\\
$

{1,2}|2\\xi(3,2)*xi(2,5) + xi(3,1)*xi(2,6) - xi(4,1)*xi(2,5) + xi(4,2)*xi(2,6)\\
$

{1,2}|3\\xi(3,6)*xi(3,1) + xi(3,5)*xi(3,2) - xi(4,1)*xi(3,5) + xi(4,2)*xi(3,6)\\
$

{1,2}|4\\(xi(4,2) + xi(3,1))*xi(4,6) - (xi(4,1) - xi(3,2))*xi(4,5)\\$

{1,2}|5\\ - (xi(3,2)*xi(1,1) - xi(3,1)*xi(1,2) - xi(4,1)*xi(2,2) + xi(4,2)*xi(2,
1) + (xi(4,1) - xi(3,2))*xi(5,5)) + (xi(4,2) + xi(3,1))*xi(5,6)\\$

{1,2}|6\\xi(3,2)*xi(2,1) - xi(3,1)*xi(2,2) + xi(4,1)*xi(1,2) - xi(4,2)*xi(1,1) -
 (xi(4,1) - xi(3,2))*xi(6,5) + (xi(4,2) + xi(3,1))*xi(6,6)\\$

{1,3}|1\\ - (xi(2,1)*xi(1,6) - xi(1,5)*xi(1,1) - xi(3,3)*xi(1,5)) + xi(4,3)*xi(1
,6)\\$

{1,3}|2\\ - (xi(2,6)*xi(2,1) - xi(2,5)*xi(1,1) - xi(3,3)*xi(2,5)) + xi(4,3)*xi(2
,6)\\$

{1,3}|3\\(xi(3,3) + xi(1,1))*xi(3,5) - xi(3,6)*xi(2,1) + xi(4,3)*xi(3,6)\\$

{1,3}|4\\(xi(4,3) - xi(2,1))*xi(4,6) + (xi(3,3) + xi(1,1))*xi(4,5)\\$

{1,3}|5\\xi(3,1)*xi(1,3) + 1 - xi(3,3)*xi(1,1) + xi(4,1)*xi(2,3) - xi(4,3)*xi(2,
1) + (xi(3,3) + xi(1,1))*xi(5,5) + (xi(4,3) - xi(2,1))*xi(5,6)\\$

{1,3}|6\\xi(3,3)*xi(2,1) - xi(3,1)*xi(2,3) + xi(4,1)*xi(1,3) - xi(4,3)*xi(1,1) +
 (xi(3,3) + xi(1,1))*xi(6,5) + (xi(4,3) - xi(2,1))*xi(6,6)\\$

{1,4}|1\\xi(2,1)*xi(1,5) + xi(1,6)*xi(1,1) + xi(3,4)*xi(1,5) + xi(4,4)*xi(1,6)\\
$

{1,4}|2\\xi(2,6)*xi(1,1) + xi(2,5)*xi(2,1) + xi(3,4)*xi(2,5) + xi(4,4)*xi(2,6)\\
$

{1,4}|3\\(xi(3,4) + xi(2,1))*xi(3,5) + xi(3,6)*xi(1,1) + xi(4,4)*xi(3,6)\\$

{1,4}|4\\(xi(4,4) + xi(1,1))*xi(4,6) + (xi(3,4) + xi(2,1))*xi(4,5)\\$

{1,4}|5\\ - (xi(3,4)*xi(1,1) - xi(3,1)*xi(1,4) - xi(4,1)*xi(2,4) + xi(4,4)*xi(2,
1) - (xi(3,4) + xi(2,1))*xi(5,5)) + (xi(4,4) + xi(1,1))*xi(5,6)\\$

{1,4}|6\\ - (xi(3,1)*xi(2,4) - 1 - xi(3,4)*xi(2,1) - xi(4,1)*xi(1,4) + xi(4,4)*
xi(1,1) - (xi(3,4) + xi(2,1))*xi(6,5)) + (xi(4,4) + xi(1,1))*xi(6,6)\\$

{1,5}|1\\xi(4,5)*xi(1,6) + xi(3,5)*xi(1,5)\\$

{1,5}|2\\xi(4,5)*xi(2,6) + xi(3,5)*xi(2,5)\\$

{1,5}|3\\xi(4,5)*xi(3,6) + xi(3,5)**2\\$

{1,5}|4\\xi(4,6)*xi(4,5) + xi(4,5)*xi(3,5)\\$

{1,5}|5\\ - (xi(3,5)*xi(1,1) - xi(3,1)*xi(1,5) - xi(4,1)*xi(2,5) + xi(4,5)*xi(2,
1) - xi(5,5)*xi(3,5)) + xi(5,6)*xi(4,5)\\$

{1,5}|6\\xi(3,5)*xi(2,1) - xi(3,1)*xi(2,5) + xi(4,1)*xi(1,5) - xi(4,5)*xi(1,1) +
 xi(6,5)*xi(3,5) + xi(6,6)*xi(4,5)\\$

{1,6}|1\\xi(4,6)*xi(1,6) + xi(3,6)*xi(1,5)\\$

{1,6}|2\\xi(4,6)*xi(2,6) + xi(3,6)*xi(2,5)\\$

{1,6}|3\\xi(4,6)*xi(3,6) + xi(3,6)*xi(3,5)\\$

{1,6}|4\\xi(4,6)**2 + xi(4,5)*xi(3,6)\\$

{1,6}|5\\ - (xi(3,6)*xi(1,1) - xi(3,1)*xi(1,6) - xi(4,1)*xi(2,6) + xi(4,6)*xi(2,
1) - xi(5,5)*xi(3,6)) + xi(5,6)*xi(4,6)\\$

{1,6}|6\\xi(3,6)*xi(2,1) - xi(3,1)*xi(2,6) + xi(4,1)*xi(1,6) - xi(4,6)*xi(1,1) +
 xi(6,5)*xi(3,6) + xi(6,6)*xi(4,6)\\$

{2,3}|1\\ - (xi(2,2)*xi(1,6) - xi(1,5)*xi(1,2) + xi(3,3)*xi(1,6)) + xi(4,3)*xi(1
,5)\\$

{2,3}|2\\ - (xi(2,6)*xi(2,2) - xi(2,5)*xi(1,2) + xi(3,3)*xi(2,6)) + xi(4,3)*xi(2
,5)\\$

{2,3}|3\\ - (xi(3,6)*xi(3,3) + xi(3,6)*xi(2,2) - xi(3,5)*xi(1,2)) + xi(4,3)*xi(3
,5)\\$

{2,3}|4\\(xi(4,3) + xi(1,2))*xi(4,5) - (xi(3,3) + xi(2,2))*xi(4,6)\\$

{2,3}|5\\ - (xi(3,3)*xi(1,2) - xi(3,2)*xi(1,3) - xi(4,2)*xi(2,3) + xi(4,3)*xi(2,
2) - (xi(4,3) + xi(1,2))*xi(5,5)) - (xi(3,3) + xi(2,2))*xi(5,6)\\$

{2,3}|6\\ - (xi(3,2)*xi(2,3) + 1 - xi(3,3)*xi(2,2) - xi(4,2)*xi(1,3) + xi(4,3)*
xi(1,2) - (xi(4,3) + xi(1,2))*xi(6,5)) - (xi(3,3) + xi(2,2))*xi(6,6)\\$

{2,4}|1\\xi(2,2)*xi(1,5) + xi(1,6)*xi(1,2) - xi(3,4)*xi(1,6) + xi(4,4)*xi(1,5)\\
$

{2,4}|2\\xi(2,6)*xi(1,2) + xi(2,5)*xi(2,2) - xi(3,4)*xi(2,6) + xi(4,4)*xi(2,5)\\
$

{2,4}|3\\ - (xi(3,6)*xi(3,4) - xi(3,6)*xi(1,2) - xi(3,5)*xi(2,2)) + xi(4,4)*xi(3
,5)\\$

{2,4}|4\\(xi(4,4) + xi(2,2))*xi(4,5) - (xi(3,4) - xi(1,2))*xi(4,6)\\$

{2,4}|5\\xi(3,2)*xi(1,4) + 1 - xi(3,4)*xi(1,2) + xi(4,2)*xi(2,4) - xi(4,4)*xi(2,
2) + (xi(4,4) + xi(2,2))*xi(5,5) - (xi(3,4) - xi(1,2))*xi(5,6)\\$

{2,4}|6\\xi(3,4)*xi(2,2) - xi(3,2)*xi(2,4) + xi(4,2)*xi(1,4) - xi(4,4)*xi(1,2) +
 (xi(4,4) + xi(2,2))*xi(6,5) - (xi(3,4) - xi(1,2))*xi(6,6)\\$

{2,5}|1\\xi(4,5)*xi(1,5) - xi(3,5)*xi(1,6)\\$

{2,5}|2\\xi(4,5)*xi(2,5) - xi(3,5)*xi(2,6)\\$

{2,5}|3\\xi(4,5)*xi(3,5) - xi(3,6)*xi(3,5)\\$

{2,5}|4\\ - xi(4,6)*xi(3,5) + xi(4,5)**2\\$

{2,5}|5\\ - (xi(3,5)*xi(1,2) - xi(3,2)*xi(1,5) - xi(4,2)*xi(2,5) + xi(4,5)*xi(2,
2) - xi(5,5)*xi(4,5)) - xi(5,6)*xi(3,5)\\$

{2,5}|6\\xi(3,5)*xi(2,2) - xi(3,2)*xi(2,5) + xi(4,2)*xi(1,5) - xi(4,5)*xi(1,2) +
 xi(6,5)*xi(4,5) - xi(6,6)*xi(3,5)\\$

{2,6}|1\\xi(4,6)*xi(1,5) - xi(3,6)*xi(1,6)\\$

{2,6}|2\\xi(4,6)*xi(2,5) - xi(3,6)*xi(2,6)\\$

{2,6}|3\\xi(4,6)*xi(3,5) - xi(3,6)**2\\$

{2,6}|4\\xi(4,6)*xi(4,5) - xi(4,6)*xi(3,6)\\$

{2,6}|5\\ - (xi(3,6)*xi(1,2) - xi(3,2)*xi(1,6) - xi(4,2)*xi(2,6) + xi(4,6)*xi(2,
2) - xi(5,5)*xi(4,6)) - xi(5,6)*xi(3,6)\\$

{2,6}|6\\xi(3,6)*xi(2,2) - xi(3,2)*xi(2,6) + xi(4,2)*xi(1,6) - xi(4,6)*xi(1,2) +
 xi(6,5)*xi(4,6) - xi(6,6)*xi(3,6)\\$

{3,4}|1\\xi(1,6)*xi(1,3) - xi(1,5)*xi(1,4) + xi(2,3)*xi(1,5) + xi(2,4)*xi(1,6)\\
$

{3,4}|2\\(xi(2,4) + xi(1,3))*xi(2,6) + (xi(2,3) - xi(1,4))*xi(2,5)\\$

{3,4}|3\\(xi(2,4) + xi(1,3))*xi(3,6) + (xi(2,3) - xi(1,4))*xi(3,5)\\$

{3,4}|4\\(xi(2,4) + xi(1,3))*xi(4,6) + (xi(2,3) - xi(1,4))*xi(4,5)\\$

{3,4}|5\\ - (xi(3,4)*xi(1,3) - xi(3,3)*xi(1,4) - xi(4,3)*xi(2,4) + xi(4,4)*xi(2,
3) - (xi(2,3) - xi(1,4))*xi(5,5)) + (xi(2,4) + xi(1,3))*xi(5,6)\\$

{3,4}|6\\xi(3,4)*xi(2,3) - xi(3,3)*xi(2,4) + xi(4,3)*xi(1,4) - xi(4,4)*xi(1,3) +
 (xi(2,3) - xi(1,4))*xi(6,5) + (xi(2,4) + xi(1,3))*xi(6,6)\\$

{3,5}|1\\xi(2,5)*xi(1,6) - xi(1,5)**2\\$

{3,5}|2\\xi(2,6)*xi(2,5) - xi(2,5)*xi(1,5)\\$

{3,5}|3\\xi(3,6)*xi(2,5) - xi(3,5)*xi(1,5)\\$

{3,5}|4\\xi(4,6)*xi(2,5) - xi(4,5)*xi(1,5)\\$

{3,5}|5\\ - (xi(3,5)*xi(1,3) - xi(3,3)*xi(1,5) - xi(4,3)*xi(2,5) + xi(4,5)*xi(2,
3) + xi(5,5)*xi(1,5)) + xi(5,6)*xi(2,5)\\$

{3,5}|6\\xi(3,5)*xi(2,3) - xi(3,3)*xi(2,5) + xi(4,3)*xi(1,5) - xi(4,5)*xi(1,3) -
 xi(6,5)*xi(1,5) + xi(6,6)*xi(2,5)\\$

{3,6}|1\\xi(2,6)*xi(1,6) - xi(1,6)*xi(1,5)\\$

{3,6}|2\\xi(2,6)**2 - xi(2,5)*xi(1,6)\\$

{3,6}|3\\xi(3,6)*xi(2,6) - xi(3,5)*xi(1,6)\\$

{3,6}|4\\xi(4,6)*xi(2,6) - xi(4,5)*xi(1,6)\\$

{3,6}|5\\ - (xi(3,6)*xi(1,3) - xi(3,3)*xi(1,6) - xi(4,3)*xi(2,6) + xi(4,6)*xi(2,
3) + xi(5,5)*xi(1,6)) + xi(5,6)*xi(2,6)\\$

{3,6}|6\\xi(3,6)*xi(2,3) - xi(3,3)*xi(2,6) + xi(4,3)*xi(1,6) - xi(4,6)*xi(1,3) -
 xi(6,5)*xi(1,6) + xi(6,6)*xi(2,6)\\$

{4,5}|1\\ - xi(2,5)*xi(1,5) - xi(1,6)*xi(1,5)\\$

{4,5}|2\\ - xi(2,6)*xi(1,5) - xi(2,5)**2\\$

{4,5}|3\\ - xi(3,6)*xi(1,5) - xi(3,5)*xi(2,5)\\$

{4,5}|4\\ - xi(4,6)*xi(1,5) - xi(4,5)*xi(2,5)\\$

{4,5}|5\\ - (xi(3,5)*xi(1,4) - xi(3,4)*xi(1,5) - xi(4,4)*xi(2,5) + xi(4,5)*xi(2,
4) + xi(5,5)*xi(2,5)) - xi(5,6)*xi(1,5)\\$

{4,5}|6\\xi(3,5)*xi(2,4) - xi(3,4)*xi(2,5) + xi(4,4)*xi(1,5) - xi(4,5)*xi(1,4) -
 xi(6,5)*xi(2,5) - xi(6,6)*xi(1,5)\\$

{4,6}|1\\ - xi(2,6)*xi(1,5) - xi(1,6)**2\\$

{4,6}|2\\ - xi(2,6)*xi(2,5) - xi(2,6)*xi(1,6)\\$

{4,6}|3\\ - xi(3,6)*xi(1,6) - xi(3,5)*xi(2,6)\\$

{4,6}|4\\ - xi(4,6)*xi(1,6) - xi(4,5)*xi(2,6)\\$

{4,6}|5\\ - (xi(3,6)*xi(1,4) - xi(3,4)*xi(1,6) - xi(4,4)*xi(2,6) + xi(4,6)*xi(2,
4) + xi(5,5)*xi(2,6)) - xi(5,6)*xi(1,6)\\$

{4,6}|6\\xi(3,6)*xi(2,4) - xi(3,4)*xi(2,6) + xi(4,4)*xi(1,6) - xi(4,6)*xi(1,4) -
 xi(6,5)*xi(2,6) - xi(6,6)*xi(1,6)\\$

{5,6}|5\\ - (xi(3,6)*xi(1,5) - xi(3,5)*xi(1,6) - xi(4,5)*xi(2,6)) - xi(4,6)*xi(2
,5)\\$

{5,6}|6\\xi(3,6)*xi(2,5) - xi(3,5)*xi(2,6) + xi(4,5)*xi(1,6) - xi(4,6)*xi(1,5)\\
$

USD$

\par Simultaneous resolution of the nonzero torsion equations and the matrix$

equation USD J^2 = -I . USD$

Suppose first USDxi(1,5) \neq 0USD.$

\\ Then one first gets$

\\ from equation USD45|1USD :$

USDxi(1,6):=-xi(2,5)USD$

\\ and then from equation USD36|2USD : USD xi(2,6)**2 = -xi(2,5)**2USD, hence$

USDxi(2,6):=0,xi(2,5):=0,xi(1,6):=0USD$

But then from equation USD35|1USD : USD xi(1,5)=0USD  a contradiction$

Hence USD xi(1,5)USD has to be USD 0 USD$

\\ USD xi(1,5):=0USD$

\par With these values, \textit{Reduce} computes again all equations.$

then, one gets from  equation USD45|2USD :$

\\ USD xi(2,5):=0USD$

\\ and from equation USD46|1USD :$

\\ USD xi(1,6):=0USD$

\par With these values, \textit{Reduce} computes again all equations.$

then, one gets from  equation USD36|2USD :$

\\ USD xi(2,6):=0USD$

Suppose now USDxi(4,6) \neq 0USD.$

\\ Then one first gets$

\\ from equation USD26|4USD : which reads USD (xi(4,5)-xi(3,6))xi(4,6)=0USD$

USDxi(4,5):=xi(3,6)USD$

\\ and then from equation USD16|4USD$

which reads USD xi(4,6)$**2 + xi(4,5)xi(3,6)=0USD$

: USD xi(4,6)**2 = -xi(3,6)**2USD, hence$

USDxi(4,6):=0,xi(3,6):=0USD$

  a contradiction$

Hence USD xi(4,6)USD has to be USD 0 USD$

\\ USD xi(4,6):=0USD$

\par With these values, \textit{Reduce} computes again all equations.$

then, one gets from  equation USD25|4USD :$

\\ USD xi(4,5):=0USD$

\\ and from equation USD26|6USD :$

\\ USD xi(3,6):=0USD$

\par With these values, \textit{Reduce} computes again all equations.$

then, one gets from  equation USD15|3USD :$

\\ USD xi(3,5):=0USD$

\par With these values, \textit{Reduce} computes again all equations.$

Then the 5x5 entry in  USD J**2 USD  is $

USD  {J**2}^5_5=xi(5,5)**2 + xi(6,5)*xi(5,6);USD\\$

Hence USD xi(5,6)xi(6,5)\neq 0 USD$

and USD xi(5,6) =(-1-xi(5,5)**2)/xi(6,5). USD$

\\ USD xi(5,6):= - (xi(5,5)**2 + 1)/xi(6,5)USD$

Moreover, the 5x6 entry in  USD J**2 USD  is $

USD  (xi(5,5) + xi(6,6))*xi(5,6);USD hence\\$

\\ USD xi(6,6):= - xi(5,5)USD$

\par With these values, \textit{Reduce} computes again all equations.$

Then the 6x1 entry in  USD J**2 USD  gives $

\\ USD xi(5,1):=( - xi(6,4)*xi(4,1) - xi(6,3)*xi(3,1) - xi(6,2)*xi(2,1) + xi(6,1
)*xi(5,5) - xi(6,1)*xi(1,1))/xi(6,5)USD$

and the 6x2 entry in  USD J**2 USD  gives $

\\ USD xi(5,2):=( - xi(6,4)*xi(4,2) - xi(6,3)*xi(3,2) + xi(6,2)*xi(5,5) - xi(6,2
)*xi(2,2) - xi(6,1)*xi(1,2))/xi(6,5)USD$

and the 6x3 entry in  USD J**2 USD  gives $

\\ USD xi(5,3):=( - xi(6,4)*xi(4,3) + xi(6,3)*xi(5,5) - xi(6,3)*xi(3,3) - xi(6,2
)*xi(2,3) - xi(6,1)*xi(1,3))/xi(6,5)USD$

and the 6x4 entry in  USD J**2 USD  gives $

\\ USD xi(5,4):=(xi(6,4)*xi(5,5) - xi(6,4)*xi(4,4) - xi(6,3)*xi(3,4) - xi(6,2)*
xi(2,4) - xi(6,1)*xi(1,4))/xi(6,5)USD$

From the trace of USD JUSD, we get:$

\\ USD xi(4,4):= - (xi(3,3) + xi(2,2) + xi(1,1))USD$

*****************************************************************$

\starline$

At the present stage, we have : $

 \par **** \4stars Now the nonzero torsion equations left are :$

Torsion equations to cancel (Latex output) : USD$

{1,2}|5\\( - (xi(3,2)*xi(1,1) - xi(3,1)*xi(1,2) - xi(4,1)*xi(2,2) + xi(4,2)*xi(2
,1))*xi(6,5) - ((xi(5,5)**2 + 1)*(xi(4,2) + xi(3,1)) + (xi(4,1) - xi(3,2))*xi(6,
5)*xi(5,5)))/xi(6,5)\\$

{1,2}|6\\xi(3,2)*xi(2,1) - xi(3,1)*xi(2,2) + xi(4,1)*xi(1,2) - xi(4,2)*xi(1,1) -
 (xi(4,2) + xi(3,1))*xi(5,5) - (xi(4,1) - xi(3,2))*xi(6,5)\\$

{1,3}|5\\((xi(3,1)*xi(1,3) + 1 - xi(3,3)*xi(1,1) + xi(4,1)*xi(2,3) - xi(4,3)*xi(
2,1))*xi(6,5) - ((xi(5,5)**2 + 1)*(xi(4,3) - xi(2,1)) - (xi(3,3) + xi(1,1))*xi(6
,5)*xi(5,5)))/xi(6,5)\\$

{1,3}|6\\xi(3,3)*xi(2,1) - xi(3,1)*xi(2,3) + xi(4,1)*xi(1,3) - xi(4,3)*xi(1,1) -
 (xi(4,3) - xi(2,1))*xi(5,5) + (xi(3,3) + xi(1,1))*xi(6,5)\\$

{1,4}|5\\(((xi(2,2) + xi(1,1))*xi(2,1) + xi(3,1)*xi(1,4) + xi(3,3)*xi(2,1) - xi(
3,4)*xi(1,1) + xi(4,1)*xi(2,4))*xi(6,5) + (xi(5,5)**2 + 1)*(xi(3,3) + xi(2,2)) +
 (xi(3,4) + xi(2,1))*xi(6,5)*xi(5,5))/xi(6,5)\\$

{1,4}|6\\xi(1,1)**2 + 1 + xi(2,2)*xi(1,1) - xi(3,1)*xi(2,4) + xi(3,3)*xi(1,1) + 
xi(3,4)*xi(2,1) + xi(4,1)*xi(1,4) + (xi(3,3) + xi(2,2))*xi(5,5) + (xi(3,4) + xi(
2,1))*xi(6,5)\\$

{2,3}|5\\( - (xi(3,3)*xi(1,2) - xi(3,2)*xi(1,3) - xi(4,2)*xi(2,3) + xi(4,3)*xi(2
,2))*xi(6,5) + (xi(5,5)**2 + 1)*(xi(3,3) + xi(2,2)) + (xi(4,3) + xi(1,2))*xi(6,5
)*xi(5,5))/xi(6,5)\\$

{2,3}|6\\ - (xi(3,2)*xi(2,3) + 1 - xi(3,3)*xi(2,2) - xi(4,2)*xi(1,3) + xi(4,3)*
xi(1,2) - (xi(3,3) + xi(2,2))*xi(5,5)) + (xi(4,3) + xi(1,2))*xi(6,5)\\$

{2,4}|5\\((xi(2,2)**2 + xi(2,2)*xi(1,1) + 1 + xi(3,2)*xi(1,4) + xi(3,3)*xi(2,2) 
- xi(3,4)*xi(1,2) + xi(4,2)*xi(2,4))*xi(6,5) + (xi(5,5)**2 + 1)*(xi(3,4) - xi(1,
2)) - (xi(3,3) + xi(1,1))*xi(6,5)*xi(5,5))/xi(6,5)\\$

{2,4}|6\\(xi(2,2) + xi(1,1))*xi(1,2) - xi(3,2)*xi(2,4) + xi(3,3)*xi(1,2) + xi(3,
4)*xi(2,2) + xi(4,2)*xi(1,4) + (xi(3,4) - xi(1,2))*xi(5,5) - (xi(3,3) + xi(1,1))
*xi(6,5)\\$

{3,4}|5\\(((xi(2,3) + xi(1,4))*xi(3,3) + (xi(2,2) + xi(1,1))*xi(2,3) - xi(3,4)*
xi(1,3) + xi(4,3)*xi(2,4))*xi(6,5) - ((xi(5,5)**2 + 1)*(xi(2,4) + xi(1,3)) - (xi
(2,3) - xi(1,4))*xi(6,5)*xi(5,5)))/xi(6,5)\\$

{3,4}|6\\ - ((xi(2,4) - xi(1,3))*xi(3,3) - (xi(2,2) + xi(1,1))*xi(1,3) - xi(3,4)
*xi(2,3) - xi(4,3)*xi(1,4) + (xi(2,4) + xi(1,3))*xi(5,5)) + (xi(2,3) - xi(1,4))*
xi(6,5)\\$

USD$

\\ \P \\$

\par The matrix USD J USD is :\\$

USD  J^1_1=xi(1,1);USD\\$

USD  J^1_2=xi(1,2);USD\\$

USD  J^1_3=xi(1,3);USD\\$

USD  J^1_4=xi(1,4);USD\\$

USD  J^1_5=0;USD\\$

USD  J^1_6=0;USD\\$

USD  J^2_1=xi(2,1);USD\\$

USD  J^2_2=xi(2,2);USD\\$

USD  J^2_3=xi(2,3);USD\\$

USD  J^2_4=xi(2,4);USD\\$

USD  J^2_5=0;USD\\$

USD  J^2_6=0;USD\\$

USD  J^3_1=xi(3,1);USD\\$

USD  J^3_2=xi(3,2);USD\\$

USD  J^3_3=xi(3,3);USD\\$

USD  J^3_4=xi(3,4);USD\\$

USD  J^3_5=0;USD\\$

USD  J^3_6=0;USD\\$

USD  J^4_1=xi(4,1);USD\\$

USD  J^4_2=xi(4,2);USD\\$

USD  J^4_3=xi(4,3);USD\\$

USD  J^4_4= - (xi(2,2) + xi(1,1) + xi(3,3));USD\\$

USD  J^4_5=0;USD\\$

USD  J^4_6=0;USD\\$

USD  J^5_1=((xi(5,5) - xi(1,1))*xi(6,1) - xi(6,2)*xi(2,1) - xi(6,3)*xi(3,1) - xi
(6,4)*xi(4,1))/xi(6,5);USD\\$

USD  J^5_2=(xi(6,2)*xi(5,5) - xi(6,2)*xi(2,2) - xi(6,1)*xi(1,2) - xi(6,3)*xi(3,2
) - xi(6,4)*xi(4,2))/xi(6,5);USD\\$

USD  J^5_3=( - (xi(6,2)*xi(2,3) + xi(6,1)*xi(1,3) - (xi(5,5) - xi(3,3))*xi(6,3) 
+ xi(6,4)*xi(4,3)))/xi(6,5);USD\\$

USD  J^5_4=((xi(2,2) + xi(1,1) + xi(3,3) + xi(5,5))*xi(6,4) - (xi(6,2)*xi(2,4) +
 xi(6,1)*xi(1,4) + xi(6,3)*xi(3,4)))/xi(6,5);USD\\$

USD  J^5_5=xi(5,5);USD\\$

USD  J^5_6=( - (xi(5,5)**2 + 1))/xi(6,5);USD\\$

USD  J^6_1=xi(6,1);USD\\$

USD  J^6_2=xi(6,2);USD\\$

USD  J^6_3=xi(6,3);USD\\$

USD  J^6_4=xi(6,4);USD\\$

USD  J^6_5=xi(6,5);USD\\$

USD  J^6_6= - xi(5,5);USD\\$

matJ:=
mat((xi(1,1),xi(1,2),xi(1,3),xi(1,4),0,0),(xi(2,1),xi(2,2),xi(2,3),xi(2,4),0,0),
(xi(3,1),xi(3,2),xi(3,3),xi(3,4),0,0),(xi(4,1),xi(4,2),xi(4,3), - (xi(2,2) + xi(
1,1) + xi(3,3)),0,0),(((xi(5,5) - xi(1,1))*xi(6,1) - xi(6,2)*xi(2,1) - xi(6,3)*
xi(3,1) - xi(6,4)*xi(4,1))/xi(6,5),(xi(6,2)*xi(5,5) - xi(6,2)*xi(2,2) - xi(6,1)*
xi(1,2) - xi(6,3)*xi(3,2) - xi(6,4)*xi(4,2))/xi(6,5),( - (xi(6,2)*xi(2,3) + xi(6
,1)*xi(1,3) - (xi(5,5) - xi(3,3))*xi(6,3) + xi(6,4)*xi(4,3)))/xi(6,5),((xi(2,2) 
+ xi(1,1) + xi(3,3) + xi(5,5))*xi(6,4) - (xi(6,2)*xi(2,4) + xi(6,1)*xi(1,4) + xi
(6,3)*xi(3,4)))/xi(6,5),xi(5,5),( - (xi(5,5)**2 + 1))/xi(6,5)),(xi(6,1),xi(6,2),
xi(6,3),xi(6,4),xi(6,5), - xi(5,5)))$

USD  {J**2}^1_1=xi(4,1)*xi(1,4) + xi(3,1)*xi(1,3) + xi(2,1)*xi(1,2) + xi(1,1)**2
;USD\\$

USD  {J**2}^1_2=xi(4,2)*xi(1,4) + xi(3,2)*xi(1,3) + xi(2,2)*xi(1,2) + xi(1,2)*xi
(1,1);USD\\$

USD  {J**2}^1_3=xi(4,3)*xi(1,4) + xi(3,3)*xi(1,3) + xi(2,3)*xi(1,2) + xi(1,3)*xi
(1,1);USD\\$

USD  {J**2}^1_4=xi(3,4)*xi(1,3) - xi(3,3)*xi(1,4) + xi(2,4)*xi(1,2) - xi(2,2)*xi
(1,4);USD\\$

USD  {J**2}^1_5=0;USD\\$

USD  {J**2}^1_6=0;USD\\$

USD  {J**2}^2_1=xi(4,1)*xi(2,4) + xi(3,1)*xi(2,3) + xi(2,2)*xi(2,1) + xi(2,1)*xi
(1,1);USD\\$

USD  {J**2}^2_2=xi(4,2)*xi(2,4) + xi(3,2)*xi(2,3) + xi(2,2)**2 + xi(2,1)*xi(1,2)
;USD\\$

USD  {J**2}^2_3=xi(4,3)*xi(2,4) + xi(3,3)*xi(2,3) + xi(2,3)*xi(2,2) + xi(2,1)*xi
(1,3);USD\\$

USD  {J**2}^2_4=xi(3,4)*xi(2,3) - xi(3,3)*xi(2,4) - xi(2,4)*xi(1,1) + xi(2,1)*xi
(1,4);USD\\$

USD  {J**2}^2_5=0;USD\\$

USD  {J**2}^2_6=0;USD\\$

USD  {J**2}^3_1=xi(4,1)*xi(3,4) + xi(3,3)*xi(3,1) + xi(3,2)*xi(2,1) + xi(3,1)*xi
(1,1);USD\\$

USD  {J**2}^3_2=xi(4,2)*xi(3,4) + xi(3,3)*xi(3,2) + xi(3,2)*xi(2,2) + xi(3,1)*xi
(1,2);USD\\$

USD  {J**2}^3_3=xi(4,3)*xi(3,4) + xi(3,3)**2 + xi(3,2)*xi(2,3) + xi(3,1)*xi(1,3)
;USD\\$

USD  {J**2}^3_4= - xi(3,4)*xi(2,2) - xi(3,4)*xi(1,1) + xi(3,2)*xi(2,4) + xi(3,1)
*xi(1,4);USD\\$

USD  {J**2}^3_5=0;USD\\$

USD  {J**2}^3_6=0;USD\\$

USD  {J**2}^4_1=xi(4,3)*xi(3,1) + xi(4,2)*xi(2,1) - xi(4,1)*xi(3,3) - xi(4,1)*xi
(2,2);USD\\$

USD  {J**2}^4_2=xi(4,3)*xi(3,2) - xi(4,2)*xi(3,3) - xi(4,2)*xi(1,1) + xi(4,1)*xi
(1,2);USD\\$

USD  {J**2}^4_3= - xi(4,3)*xi(2,2) - xi(4,3)*xi(1,1) + xi(4,2)*xi(2,3) + xi(4,1)
*xi(1,3);USD\\$

USD  {J**2}^4_4=xi(4,3)*xi(3,4) + xi(4,2)*xi(2,4) + xi(4,1)*xi(1,4) + xi(3,3)**2
 + 2*xi(3,3)*xi(2,2) + 2*xi(3,3)*xi(1,1) + xi(2,2)**2 + 2*xi(2,2)*xi(1,1) + xi(1
,1)**2;USD\\$

USD  {J**2}^4_5=0;USD\\$

USD  {J**2}^4_6=0;USD\\$

USD  {J**2}^5_1=( - xi(6,4)*xi(4,3)*xi(3,1) - xi(6,4)*xi(4,2)*xi(2,1) + xi(6,4)*
xi(4,1)*xi(3,3) + xi(6,4)*xi(4,1)*xi(2,2) - xi(6,3)*xi(4,1)*xi(3,4) - xi(6,3)*xi
(3,3)*xi(3,1) - xi(6,3)*xi(3,2)*xi(2,1) - xi(6,3)*xi(3,1)*xi(1,1) - xi(6,2)*xi(4
,1)*xi(2,4) - xi(6,2)*xi(3,1)*xi(2,3) - xi(6,2)*xi(2,2)*xi(2,1) - xi(6,2)*xi(2,1
)*xi(1,1) - xi(6,1)*xi(4,1)*xi(1,4) - xi(6,1)*xi(3,1)*xi(1,3) - xi(6,1)*xi(2,1)*
xi(1,2) - xi(6,1)*xi(1,1)**2 - xi(6,1))/xi(6,5);USD\\$

USD  {J**2}^5_2=( - xi(6,4)*xi(4,3)*xi(3,2) + xi(6,4)*xi(4,2)*xi(3,3) + xi(6,4)*
xi(4,2)*xi(1,1) - xi(6,4)*xi(4,1)*xi(1,2) - xi(6,3)*xi(4,2)*xi(3,4) - xi(6,3)*xi
(3,3)*xi(3,2) - xi(6,3)*xi(3,2)*xi(2,2) - xi(6,3)*xi(3,1)*xi(1,2) - xi(6,2)*xi(4
,2)*xi(2,4) - xi(6,2)*xi(3,2)*xi(2,3) - xi(6,2)*xi(2,2)**2 - xi(6,2)*xi(2,1)*xi(
1,2) - xi(6,2) - xi(6,1)*xi(4,2)*xi(1,4) - xi(6,1)*xi(3,2)*xi(1,3) - xi(6,1)*xi(
2,2)*xi(1,2) - xi(6,1)*xi(1,2)*xi(1,1))/xi(6,5);USD\\$

USD  {J**2}^5_3=(xi(6,4)*xi(4,3)*xi(2,2) + xi(6,4)*xi(4,3)*xi(1,1) - xi(6,4)*xi(
4,2)*xi(2,3) - xi(6,4)*xi(4,1)*xi(1,3) - xi(6,3)*xi(4,3)*xi(3,4) - xi(6,3)*xi(3,
3)**2 - xi(6,3)*xi(3,2)*xi(2,3) - xi(6,3)*xi(3,1)*xi(1,3) - xi(6,3) - xi(6,2)*xi
(4,3)*xi(2,4) - xi(6,2)*xi(3,3)*xi(2,3) - xi(6,2)*xi(2,3)*xi(2,2) - xi(6,2)*xi(2
,1)*xi(1,3) - xi(6,1)*xi(4,3)*xi(1,4) - xi(6,1)*xi(3,3)*xi(1,3) - xi(6,1)*xi(2,3
)*xi(1,2) - xi(6,1)*xi(1,3)*xi(1,1))/xi(6,5);USD\\$

USD  {J**2}^5_4=( - xi(6,4)*xi(4,3)*xi(3,4) - xi(6,4)*xi(4,2)*xi(2,4) - xi(6,4)*
xi(4,1)*xi(1,4) - xi(6,4)*xi(3,3)**2 - 2*xi(6,4)*xi(3,3)*xi(2,2) - 2*xi(6,4)*xi(
3,3)*xi(1,1) - xi(6,4)*xi(2,2)**2 - 2*xi(6,4)*xi(2,2)*xi(1,1) - xi(6,4)*xi(1,1)
**2 - xi(6,4) + xi(6,3)*xi(3,4)*xi(2,2) + xi(6,3)*xi(3,4)*xi(1,1) - xi(6,3)*xi(3
,2)*xi(2,4) - xi(6,3)*xi(3,1)*xi(1,4) - xi(6,2)*xi(3,4)*xi(2,3) + xi(6,2)*xi(3,3
)*xi(2,4) + xi(6,2)*xi(2,4)*xi(1,1) - xi(6,2)*xi(2,1)*xi(1,4) - xi(6,1)*xi(3,4)*
xi(1,3) + xi(6,1)*xi(3,3)*xi(1,4) - xi(6,1)*xi(2,4)*xi(1,2) + xi(6,1)*xi(2,2)*xi
(1,4))/xi(6,5);USD\\$

USD  {J**2}^5_5=-1;USD\\$

USD  {J**2}^5_6=0;USD\\$

USD  {J**2}^6_1=0;USD\\$

USD  {J**2}^6_2=0;USD\\$

USD  {J**2}^6_3=0;USD\\$

USD  {J**2}^6_4=0;USD\\$

USD  {J**2}^6_5=0;USD\\$

USD  {J**2}^6_6=-1;USD\\$

\\$ det J:= - xi(4,3)*xi(3,4)*xi(2,2)*xi(1,1) + xi(4,3)*xi(3,4)*xi(2,1)*xi(1,2) 
+ xi(4,3)*xi(3,2)*xi(2,4)*xi(1,1) - xi(4,3)*xi(3,2)*xi(2,1)*xi(1,4) - xi(4,3)*xi
(3,1)*xi(2,4)*xi(1,2) + xi(4,3)*xi(3,1)*xi(2,2)*xi(1,4) + xi(4,2)*xi(3,4)*xi(2,3
)*xi(1,1) - xi(4,2)*xi(3,4)*xi(2,1)*xi(1,3) - xi(4,2)*xi(3,3)*xi(2,4)*xi(1,1) + 
xi(4,2)*xi(3,3)*xi(2,1)*xi(1,4) + xi(4,2)*xi(3,1)*xi(2,4)*xi(1,3) - xi(4,2)*xi(3
,1)*xi(2,3)*xi(1,4) - xi(4,1)*xi(3,4)*xi(2,3)*xi(1,2) + xi(4,1)*xi(3,4)*xi(2,2)*
xi(1,3) + xi(4,1)*xi(3,3)*xi(2,4)*xi(1,2) - xi(4,1)*xi(3,3)*xi(2,2)*xi(1,4) - xi
(4,1)*xi(3,2)*xi(2,4)*xi(1,3) + xi(4,1)*xi(3,2)*xi(2,3)*xi(1,4) - xi(3,3)**2*xi(
2,2)*xi(1,1) + xi(3,3)**2*xi(2,1)*xi(1,2) + xi(3,3)*xi(3,2)*xi(2,3)*xi(1,1) - xi
(3,3)*xi(3,2)*xi(2,1)*xi(1,3) - xi(3,3)*xi(3,1)*xi(2,3)*xi(1,2) + xi(3,3)*xi(3,1
)*xi(2,2)*xi(1,3) - xi(3,3)*xi(2,2)**2*xi(1,1) + xi(3,3)*xi(2,2)*xi(2,1)*xi(1,2)
 - xi(3,3)*xi(2,2)*xi(1,1)**2 + xi(3,3)*xi(2,1)*xi(1,2)*xi(1,1) + xi(3,2)*xi(2,3
)*xi(2,2)*xi(1,1) + xi(3,2)*xi(2,3)*xi(1,1)**2 - xi(3,2)*xi(2,2)*xi(2,1)*xi(1,3)
 - xi(3,2)*xi(2,1)*xi(1,3)*xi(1,1) - xi(3,1)*xi(2,3)*xi(2,2)*xi(1,2) - xi(3,1)*
xi(2,3)*xi(1,2)*xi(1,1) + xi(3,1)*xi(2,2)**2*xi(1,3) + xi(3,1)*xi(2,2)*xi(1,3)*
xi(1,1)$

Trace J:=0$

*****************************************************************$

\starline$

Note that USD xi(5,j) USD is a linear combination of the USD xi(6,j)USD's$

USD 1\leq j \leq 4 USD$

hence vanish if all USD xi(6,j)USD's do$

\par Now we'll use equivalence by automorphisms.$

All automorphisms of$

USD {\mathcal{G}}_{6,m5}USD$

are  of the following  form  :$

\\$

USDUSD \Phi = \begin{pmatrix}$

b(1,1)&$

b(2,1)*u&$

b(1,3)&$

 - b(2,3)*u&$

0&$

0\\$

b(2,1)&$

 - b(1,1)*u&$

b(2,3)&$

b(1,3)*u&$

0&$

0\\$

b(3,1)&$

 - b(4,1)*u&$

b(3,3)&$

b(4,3)*u&$

0&$

0\\$

b(4,1)&$

b(3,1)*u&$

b(4,3)&$

 - b(3,3)*u&$

0&$

0\\$

b(5,1)&$

b(5,2)&$

b(5,3)&$

b(5,4)&$

b(4,3)*b(2,1) - b(4,1)*b(2,3) + b(3,3)*b(1,1) - b(3,1)*b(1,3)&$

u*(b(4,3)*b(1,1) - b(4,1)*b(1,3) - b(3,3)*b(2,1) + b(3,1)*b(2,3))\\$

b(6,1)&$

b(6,2)&$

b(6,3)&$

b(6,4)&$

b(4,3)*b(1,1) - b(4,1)*b(1,3) - b(3,3)*b(2,1) + b(3,1)*b(2,3)&$

u*( - b(4,3)*b(2,1) + b(4,1)*b(2,3) - b(3,3)*b(1,1) + b(3,1)*b(1,3))
\end{pmatrix}USDUSD$

where USDu**2=1 USD\\$

****Take the following values :$

USD b(5,j):=$

\frac {-xi(6,j)+ \sum_{k=1}^{4} b(6,k)*xi(k,j) -xi(6,6)*b(6,j)}{xi(6,5)} USD$

and :$

\\$

USDUSD \Phi = \begin{pmatrix}$

1&$

0&$

0&$

0&$

0&$

0\\$

0&$

1&$

0&$

0&$

0&$

0\\$

0&$

0&$

1&$

0&$

0&$

0\\$

0&$

0&$

0&$

1&$

0&$

0\\$

b(5,1)&$

b(5,2)&$

b(5,3)&$

b(5,4)&$

1&$

0\\$

b(6,1)&$

b(6,2)&$

b(6,3)&$

b(6,4)&$

0&$

1\end{pmatrix}USDUSD$

USD det \Phi:=1USD$

\\ Then USD J2:=\Phi^{-1}*J*\Phi USD has entries :$

\\USD J2(1,1):=xi(1,1)USD\\$

\\USD J2(1,2):=xi(1,2)USD\\$

\\USD J2(1,3):=xi(1,3)USD\\$

\\USD J2(1,4):=xi(1,4)USD\\$

\\USD J2(1,5):=0USD\\$

\\USD J2(1,6):=0USD\\$

\\USD J2(2,1):=xi(2,1)USD\\$

\\USD J2(2,2):=xi(2,2)USD\\$

\\USD J2(2,3):=xi(2,3)USD\\$

\\USD J2(2,4):=xi(2,4)USD\\$

\\USD J2(2,5):=0USD\\$

\\USD J2(2,6):=0USD\\$

\\USD J2(3,1):=xi(3,1)USD\\$

\\USD J2(3,2):=xi(3,2)USD\\$

\\USD J2(3,3):=xi(3,3)USD\\$

\\USD J2(3,4):=xi(3,4)USD\\$

\\USD J2(3,5):=0USD\\$

\\USD J2(3,6):=0USD\\$

\\USD J2(4,1):=xi(4,1)USD\\$

\\USD J2(4,2):=xi(4,2)USD\\$

\\USD J2(4,3):=xi(4,3)USD\\$

\\USD J2(4,4):= - (xi(2,2) + xi(1,1) + xi(3,3))USD\\$

\\USD J2(4,5):=0USD\\$

\\USD J2(4,6):=0USD\\$

\\USD J2(5,1):=((xi(5,5) - xi(1,1))*xi(6,1) - xi(6,2)*xi(2,1) - xi(6,3)*xi(3,1) 
- xi(6,4)*xi(4,1) + (xi(5,5) - xi(1,1))*b(5,1)*xi(6,5) - b(5,2)*xi(6,5)*xi(2,1) 
- b(5,3)*xi(6,5)*xi(3,1) - b(5,4)*xi(6,5)*xi(4,1) - (xi(5,5)**2 + 1)*b(6,1))/xi(
6,5)USD\\$

\\USD J2(5,2):=(xi(6,2)*xi(5,5) - xi(6,2)*xi(2,2) - xi(6,1)*xi(1,2) - xi(6,3)*xi
(3,2) - xi(6,4)*xi(4,2) - b(5,1)*xi(6,5)*xi(1,2) + (xi(5,5) - xi(2,2))*b(5,2)*xi
(6,5) - b(5,3)*xi(6,5)*xi(3,2) - b(5,4)*xi(6,5)*xi(4,2) - (xi(5,5)**2 + 1)*b(6,2
))/xi(6,5)USD\\$

\\USD J2(5,3):=( - (xi(6,2)*xi(2,3) + xi(6,1)*xi(1,3) - (xi(5,5) - xi(3,3))*xi(6
,3) + xi(6,4)*xi(4,3) + b(5,1)*xi(6,5)*xi(1,3) + b(5,2)*xi(6,5)*xi(2,3) - (xi(5,
5) - xi(3,3))*b(5,3)*xi(6,5) + b(5,4)*xi(6,5)*xi(4,3) + (xi(5,5)**2 + 1)*b(6,3))
)/xi(6,5)USD\\$

\\USD J2(5,4):=((xi(2,2) + xi(1,1) + xi(3,3) + xi(5,5))*xi(6,4) - (xi(6,2)*xi(2,
4) + xi(6,1)*xi(1,4) + xi(6,3)*xi(3,4)) - b(5,1)*xi(6,5)*xi(1,4) - b(5,2)*xi(6,5
)*xi(2,4) - b(5,3)*xi(6,5)*xi(3,4) + (xi(2,2) + xi(1,1) + xi(3,3) + xi(5,5))*b(5
,4)*xi(6,5) - (xi(5,5)**2 + 1)*b(6,4))/xi(6,5)USD\\$

\\USD J2(5,5):=xi(5,5)USD\\$

\\USD J2(5,6):=( - (xi(5,5)**2 + 1))/xi(6,5)USD\\$

\\USD J2(6,1):=b(5,1)*xi(6,5) + xi(6,1) - (xi(5,5) + xi(1,1))*b(6,1) - b(6,2)*xi
(2,1) - b(6,3)*xi(3,1) - b(6,4)*xi(4,1)USD\\$

\\USD J2(6,2):=b(5,2)*xi(6,5) + xi(6,2) - b(6,1)*xi(1,2) - (xi(5,5) + xi(2,2))*b
(6,2) - b(6,3)*xi(3,2) - b(6,4)*xi(4,2)USD\\$

\\USD J2(6,3):=b(5,3)*xi(6,5) + xi(6,3) - b(6,1)*xi(1,3) - b(6,2)*xi(2,3) - (xi(
5,5) + xi(3,3))*b(6,3) - b(6,4)*xi(4,3)USD\\$

\\USD J2(6,4):=b(5,4)*xi(6,5) + xi(6,4) - b(6,1)*xi(1,4) - b(6,2)*xi(2,4) - b(6,
3)*xi(3,4) + (xi(2,2) + xi(1,1) + xi(3,3) - xi(5,5))*b(6,4)USD\\$

\\USD J2(6,5):=xi(6,5)USD\\$

\\USD J2(6,6):= - xi(5,5)USD\\$

USD det \Phi:=1USD$

Note that in USD J2 USD the 5xj and 6xj terms vanish $

(USD1 \leqslant j \leqslant 4)USD from the remaining equations as it must be$

**** \4stars Hence, we are led to the case where$

USD xi(5,j)=xi(6,j) = 0 \forall j \, 1 \leqslant j \leqslant 4 USD$

clear USD b(1,1),b(1,3),b(2,1),b(2,3),b(3,1),b(4,1),b(3,3),b(4,3),u USD$

xi(5,1):=0$

xi(5,2):=0$

xi(5,3):=0$

xi(5,4):=0$

xi(6,1):=0$

xi(6,2):=0$

xi(6,3):=0$

xi(6,4):=0$

*****************************************************************$

\starline$

**** Until this stage, there was no difference with case 1$

*****************************************************************$

\starline$

**** In the present  case 2$

We now suppose USD B=C=0 USD$

\\ USD xi(1,3):=0USD$

\\ USD xi(1,4):=0USD$

\\ USD xi(2,3):=0USD$

\\ USD xi(2,4):=0USD$

\\ USD xi(3,1):=0USD$

\\ USD xi(3,2):=0USD$

\\ USD xi(4,1):=0USD$

\\ USD xi(4,2):=0USD$

Then the 1x1 entry in  USD J**2 USD  gives that xi(1,2)xi(2,1) and \neq 0$

\\ USD xi(1,2):= - (xi(1,1)**2 + 1)/xi(2,1)USD$

The 1x2 entry in  USD J**2 USD  gives$

\\ USD xi(2,2):= - xi(1,1)USD$

**** Then the 3x3 entry in  USD J**2 USD  gives that xi(3,4)xi(4,3) and \neq 0$

\\ USD xi(3,4):= - (xi(3,3)**2 + 1)/xi(4,3)USD$

*****************************************************************$

\starline$

Now the matrix  USD J USD is$

USD  J^1_1=xi(1,1);USD\\$

USD  J^1_2=( - (xi(1,1)**2 + 1))/xi(2,1);USD\\$

USD  J^1_3=0;USD\\$

USD  J^1_4=0;USD\\$

USD  J^1_5=0;USD\\$

USD  J^1_6=0;USD\\$

USD  J^2_1=xi(2,1);USD\\$

USD  J^2_2= - xi(1,1);USD\\$

USD  J^2_3=0;USD\\$

USD  J^2_4=0;USD\\$

USD  J^2_5=0;USD\\$

USD  J^2_6=0;USD\\$

USD  J^3_1=0;USD\\$

USD  J^3_2=0;USD\\$

USD  J^3_3=xi(3,3);USD\\$

USD  J^3_4=( - (xi(3,3)**2 + 1))/xi(4,3);USD\\$

USD  J^3_5=0;USD\\$

USD  J^3_6=0;USD\\$

USD  J^4_1=0;USD\\$

USD  J^4_2=0;USD\\$

USD  J^4_3=xi(4,3);USD\\$

USD  J^4_4= - xi(3,3);USD\\$

USD  J^4_5=0;USD\\$

USD  J^4_6=0;USD\\$

USD  J^5_1=0;USD\\$

USD  J^5_2=0;USD\\$

USD  J^5_3=0;USD\\$

USD  J^5_4=0;USD\\$

USD  J^5_5=xi(5,5);USD\\$

USD  J^5_6=( - (xi(5,5)**2 + 1))/xi(6,5);USD\\$

USD  J^6_1=0;USD\\$

USD  J^6_2=0;USD\\$

USD  J^6_3=0;USD\\$

USD  J^6_4=0;USD\\$

USD  J^6_5=xi(6,5);USD\\$

USD  J^6_6= - xi(5,5);USD\\$

*****************************************************************$

\starline$

By equivalence, we can suppose USD xi(1,1):=0 , xi(3,3):=0 USD$

\\ USD xi(1,1):=0USD$

\\ USD xi(3,3):=0USD$

**** \4stars In the present$

case 2.2$

Suppose now USDxi(5,5) \neq 0USD.$

\\ Then one first gets$

\\ from equation USD13|6USD : which reads USD (xi(4,3)-xi(2,1))xi(5,5)=0USD$

USDxi(4,3):=xi(2,1)USD$

\\ USD xi(4,3):=xi(2,1)USD$

\par With these values, \textit{Reduce} computes again all equations.$

\\ Then equation USD13|5USD  is USD -xi(2,1)**2+1=0USD$

\\ USD xi(2,1)**2:=1USD$

\par Now the nonzero torsion equations left are :$

Torsion equations to cancel (Latex output) : USD$

USD$

\\ \P \\$

\par The matrix USD J USD is :\\$

\\$

USDUSD J = \begin{pmatrix}$

0&$

( - 1)/xi(2,1)&$

0&$

0&$

0&$

0\\$

xi(2,1)&$

0&$

0&$

0&$

0&$

0\\$

0&$

0&$

0&$

( - 1)/xi(2,1)&$

0&$

0\\$

0&$

0&$

xi(2,1)&$

0&$

0&$

0\\$

0&$

0&$

0&$

0&$

xi(5,5)&$

( - (xi(5,5)**2 + 1))/xi(6,5)\\$

0&$

0&$

0&$

0&$

xi(6,5)&$

 - xi(5,5)\end{pmatrix}USDUSD$

USDUSD J^2 = \begin{pmatrix}$

-1&$

0&$

0&$

0&$

0&$

0\\$

0&$

-1&$

0&$

0&$

0&$

0\\$

0&$

0&$

-1&$

0&$

0&$

0\\$

0&$

0&$

0&$

-1&$

0&$

0\\$

0&$

0&$

0&$

0&$

-1&$

0\\$

0&$

0&$

0&$

0&$

0&$

-1\end{pmatrix}USDUSD$

\\USD det J:=1 USD$

USD Trace J:=0 USD$

\par Commutation relations of USD \mathfrak{m} : USD$

\P$

\par Now we check if the condition USD [Jx,y]= J[x,y] USD is satisfied$

USD\forall x,y \in {\mathcal{G}}_{6,m5},USD$

\textit{i.e.} if USD{\mathcal{G}}_{6,m5}USD$

is a \textit{complex} algebra.$

\\USD J[x_j,x_k] \neq [Jx_j,x_k] USD
in the following cases{{{1,3}, - x(6)*(xi(6,5) + xi(2,1)) - xi(5,5)*x(5)},
{{1,4},
xi(5,5)*x(6) + (x(5)*(xi(6,5)*xi(2,1) + xi(5,5)**2 + 1))/xi(6,5)},
{{2,3},
 - xi(5,5)*x(6) - (x(5)*(xi(6,5) + xi(5,5)**2*xi(2,1) + xi(2,1)))/(xi(6,5)*xi(2,
1))},
{{2,4},
 - (x(6)*(xi(6,5)*xi(2,1) + 1))/xi(2,1) - xi(5,5)*x(5)},
{{3,1},x(6)*(xi(6,5) - xi(2,1)) + xi(5,5)*x(5)},
{{3,2},
xi(5,5)*x(6) + (x(5)*( - xi(6,5)*xi(2,1) + xi(5,5)**2 + 1))/xi(6,5)},
{{4,1},
 - xi(5,5)*x(6) + (x(5)*(xi(6,5) - xi(5,5)**2*xi(2,1) - xi(2,1)))/(xi(6,5)*xi(2,
1))},
{{4,2},
(x(6)*(xi(6,5)*xi(2,1) - 1))/xi(2,1) + xi(5,5)*x(5)}}$

\end{document}$