%This is "program10_1.red"   (formerly "structcomplM10case1.tex") :
%computes the complex structures on the Lie algebra G_{M,10}
%in the case  xi(4,3) NEQ xi(2,1)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%  OUTPUT   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 off echo$     off nat$
 OUT  "rprogram10_1.tex"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 ON RAT$
 OFF MSG$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%% Loading the commutation relations file  %%%%%%%%%%%%%%
%DIM:=(dimension de l'algebre)$
DIM:= 6$
  in "6nilp/m.10"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%ECRITURE DES RELATIONS DE COMMUTATION EN TEX
WRITE "\documentclass{article}"$
WRITE "\usepackage{amsmath,amssymb}"$
write "\setlength{\textwidth}{15cm}"$
write "\setlength{\textheight}{25cm}"$
write "\setlength{\voffset}{-2cm}" $
write " \setlength{\oddsidemargin}{0in}"$
write " \setlength{\evensidemargin}{0.36in}"$
WRITE "\sloppy"$
WRITE "\begin{document}"$
%WRITE "This output from the file \texttt{structcomplM10case1.tex}.\\"$
WRITE "This output from the file \texttt{program10_1.red}.\\"$
WRITE "Computation of all complex    structures on the real Lie Algebra"$
write "USD {\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}.USD"$
%WRITE "textbf{up to equivalence}"$
write "\\ Case USD xi(4,3) \neq xi(2,1) USD"$
WRITE "\smallskip  \par "$
WRITE "Commutation relations for"$
write "USD {\mathcal{G}}_{", dim,",", PART(REFALGTEX,1), "}:USD\\"$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO X_(i,j):=X(i)*x(j)$
%FOR j:=1:DIM DO X(j):=MKID(x_,j)$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO
    IF X_(i,j) NEQ 0 THEN WRITE
 %       "USD[x_",i,",x_",j,"]=", X_(i,j),"USD;"$
        "USD[x(",i,"),x(",j,")]=", X_(i,j),"USD;"$
WRITE "\P"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 operator x,f,A$
 noncom x,f$
 for all j let
  F(X(j))=FOR j1:=1:DIM SUM xi(j1,j)*X(j1)$
  let f(0)=0$
 for all j let f(-x(j))=-f(x(j))$
 FOR ALL J,U SUCH THAT NUMBERP(U) LET f(U*X(j))=U*f(X(j))$
 FOR ALL J,U,V SUCH THAT NUMBERP(U) AND NUMBERP(V) LET f(U*V*X(j))=U*V*f(X(j))$
 FOR ALL J,U,V SUCH THAT NUMBERP(U) AND NUMBERP(V) LET f(-U*V*X(j))=-U*V*f(X(j))$
 FOR ALL s,i,j,k,l
 LET f(X(s) *xi(i,j)*xi(k,l))=xi(i,j)*xi(k,l)*f(X(s))$
 FOR ALL s,i,j,k,l
 LET f(xi(i,j)*xi(k,l)*X(s))=xi(i,j)*xi(k,l)*f(X(s))$
 FOR ALL s,i,j,k,l
 LET f(-X(s) *xi(i,j)*xi(k,l))=-xi(i,j)*xi(k,l)*f(X(s))$
 FOR ALL s,i,j
 LET f(X(s) *xi(i,j))=(xi(i,j))*f(X(s))$
 FOR ALL s,i,j
 LET f(-X(s) *xi(i,j))=-(xi(i,j))*f(X(s))$
 FOR ALL s,i,j
 LET f(-X(s) *xi(i,j)**2)=(-xi(i,j)**2)*f(X(s))$
 FOR ALL s,i,j
 LET f(X(s) *xi(i,j)**2)=(xi(i,j)**2)*f(X(s))$
 FOR ALL s,i,j,k,l
 LET f(-xi(i,j)*xi(k,l)*X(s))=-xi(i,j)*xi(k,l)*f(X(s))$
 FOR ALL J,U SUCH THAT NUMBERP(U) LET f(-U*X(j))=-U*f(X(j))$
 FOR ALL A,B LET F(A+B)=F(A)+F(B)$
 %LIGNE A MODIFIER POUR LA LINEARITE SUR LE PARAMETRE CONTINU L
 for all j let f(L*x(j))=L*f(x(j))$
 for all j let f(-L*x(j))=-L*f(x(j))$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%% collecting the torsion equations
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 for all j,k let
  temp110(j,k) = x(j)*F(x(k))$
 for all j,k let
  temp120(j,k) = F(temp110(j,k))$
 for all j,k let
  temp101(j,k) = F(x(j))*x(k)$
 for all j,k let
  temp102(j,k) = F(temp101(j,k))$
 for all j,k let
  temp2(j,k) =    F(x(j))* F(x(k)) $
 for all j,k let  temp0(j,k) =  x(j)*x(k)$
 for all j,k let
  A(j,k)=  -temp2(j,k) + temp0(j,k) + temp120(j,k) + temp102(j,k);

COLLECT_TORSION:=FOR j1:=1:DIM-1 JOIN
   FOR j2:=j1+1:DIM JOIN
   IF A(j1,j2) NEQ 0 THEN
   {{{j1,j2},A(j1,j2)}}
   ELSE {}$

%IF LENGTH(COLLECT_TORSION) NEQ 0 THEN WRITE "Nonzero torsion
%   in the following cases :",COLLECT_TORSION

IF LENGTH(COLLECT_TORSION) NEQ 0 THEN WRITE "Nonzero torsion"
ELSE WRITE "Zero torsion"$

write "\par"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off exp$
on factor$

%la liste des équations non nulles
%write "list of the nonzero torsion equations"$
COLLECT_EQ:=FOR j1:=1:LENGTH(COLLECT_TORSION) JOIN
   FOR j2:=1:DIM JOIN
   IF V(j2)*PART(PART(COLLECT_TORSION,j1),2) NEQ 0 THEN
   {{{PART(PART(COLLECT_TORSION,j1),1),j2}, V(j2)*PART(PART(COLLECT_TORSION,j1),2)} }
    ELSE {}$

%COMMENT
% WRITE "Torsion equations to cancel (Reduce output) : \\", COLLECT_EQ$

 WRITE "\\Torsion equations to cancel (Latex output) : \\USD"$
 for each A in COLLECT_EQ do
        if PART(A,2) neq 0 then
        <<write PART(PART(A,1),1),"|", PART(PART(A,1),2),"\\",PART(A,2),"\\">>   $
        %<<write "\\", A,"\\">>   $
  write "USD"$


 off factor$
 on exp$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% writing the torsion equations in the specific file
out "matJ10_1.red"$
WRITE "% This file contains the REDUCE data for the general complex structures"$
WRITE "% and the torsion equation to be canceled"$
write "% on the Lie algebra g_{", dim, ",", PART(REFALGTEX,1), "}"$
write "% in the case  xi(4,3) NEQ xi(2,1)"$
WRITE "matrix J(6,6)"$
%WRITE "J:=",MATJ$
WRITE "% Writing the torsion equations in Reduce format:"$

FOR i:=1:DIM DO FOR k:=1:DIM DO  << xi(i,k) := J(i,k)>>$
write "COLLECT_EQ:=",COLLECT_EQ$

for i:=1:DIM do for k:=1:DIM do <<CLEAR xi(i,k)>>$
clear J$

 out "rprogram10_1.tex"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% computing the complex structures
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%write "*****************************************************************"$
write "\\ \starline"$
write "\par Simultaneous resolution of the nonzero torsion equations and the matrix"$
write "equation USD J^2 = -I . USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Suppose first USDxi(1,5) \neq 0USD."$
write "\\ Then one first gets"$
write "\\ from equation USD45|1USD :"$
write "USD xi(2,5):=-xi(1,6)USD"$
write "But then equation write USD35|1USD reads :"$
write "USD -xi(2,5)**2 -xi(1,5)**2 =0USD"$
write "hence, one gets from  equation USD35|1 : xi(1,5)=0 USD, a contradiction "$
write "Hence USD xi(1,5)USD has to be USD 0 USD"$
xi(1,5):=0$
write "\\ USD xi(1,5):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "then, one gets from  equation USD45|2USD :"$
xi(2,5):=0$
write "\\ USD xi(2,5):=", ws,"USD"$
write "\\ and from equation USD46|1USD :"$
xi(1,6):=0$
write "\\ USD xi(1,6):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "then, one gets from  equation USD36|2USD :"$
xi(2,6):=0$
write "\\ USD xi(2,6):=", ws,"USD"$
write "\\ and from equation USD23|1USD :"$
xi(1,3):=0$
write "\\ USD xi(1,3):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "then, one gets from  equation USD13|2USD :"$
xi(2,3):=0$
write "\\ USD xi(2,3):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Suppose now USDxi(4,5) \neq 0USD."$
write "\\ Then one first gets"$
write "\\ from equation USD45|4USD :"$
write "USD xi(4,6):=-xi(3,5)USD"$
write "But then equation write USD25|4USD reads :"$
write "USD xi(3,5)**2 + xi(4,5)**2 =0USD"$
write "hence, one gets from  equation USD25|4 : xi(4,5)=0 USD, a contradiction "$
write "Hence USD xi(4,5)USD has to be USD 0 USD"$
xi(4,5):=0$
write "\\ USD xi(4,5):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "then, one gets from  equation USD15|3USD :"$
xi(3,5):=0$
write "\\ USD xi(3,5):=", ws,"USD"$
write "\\ and from equation USD16|4USD :"$
xi(4,6):=0$
write "\\ USD xi(4,6):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "then, one gets from  equation USD26|3USD :"$
xi(3,6):=0$
write "\\ USD xi(3,6):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 5x5 entry in  USD J**2 USD  is "$
write "USD  {J**2}^5_5=xi(5,5)**2 + xi(6,5)*xi(5,6);USD\\"$
write "Hence USD xi(5,6)xi(6,5)\neq 0 USD"$
write "and USD xi(5,6) =(-1-xi(5,5)**2)/xi(6,5). USD"$
xi(5,6):= (-1-xi(5,5)**2)/xi(6,5)$
write "\\ USD xi(5,6):=", ws,"USD"$
write "Moreover, the 5x6 entry in  USD J**2 USD  is "$
write "USD  (xi(5,5) + xi(6,6))*xi(5,6);USD hence\\"$
xi(6,6):= -xi(5,5)$
write "\\ USD xi(6,6):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then the 4x4 entry in  USD J**2 USD  is "$
write "USD  {J**2}^4_4=xi(4,4)**2 + xi(4,3)*xi(3,4);USD\\"$
write "Hence USD xi(4,3)xi(3,4)\neq 0 USD"$
write "and USD xi(3,4) =(-1-xi(4,4)**2)/xi(4,3). USD"$
xi(3,4) :=(-1-xi(4,4)**2)/xi(4,3)$
write "\\ USD xi(3,4):=", ws,"USD"$
write "Moreover, the 4x3 entry in  USD J**2 USD  is "$
write "USD  (xi(4,4) + xi(3,3))*xi(4,3);USD hence\\"$
xi(4,4):= -xi(3,3)$
write "\\ USD xi(4,4):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "then, one gets from  equation USD12|4USD :"$
xi(2,2):=-xi(1,1)$
write "\\ USD xi(2,2):=", ws,"USD"$
write "\\ and from equation USD14|4USD :"$
xi(2,4):=0$
write "\\ USD xi(2,4):=", ws,"USD"$
write "\\ and from equation USD24|4USD :"$
xi(1,4):=0$
write "\\ USD xi(1,4):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 1x1 entry in  USD J**2 USD  is "$
write "USD  {J**2}^1_1=xi(1,1)**2 + xi(2,1)*xi(1,2);USD\\"$
write "Hence USD xi(2,1)xi(1,2)\neq 0 USD"$
write "and USD xi(1,2) =(-1-xi(1,1)**2)/xi(2,1). USD"$
xi(1,2) :=(-1-xi(1,1)**2)/xi(2,1)$
write "\\ USD xi(1,2):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 4x1 entry in  USD J**2 USD  gives "$
%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
%(1,1);USD\\$
xi(4,2):=-(xi(4,3)*xi(3,1)
%+ xi(4,2)*xi(2,1)
- xi(4,1)*xi(3,3) + xi(4,1)*xi(1,1))/xi(2,1)$
write "\\ USD xi(4,2):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 5x3 entry in  USD J**2 USD  gives "$
%USD  {J**2}^5_3=(xi(6,5)*xi(5,5)*xi(5,3) + xi(6,5)*xi(5,4)*xi(4,3) + xi(6,5)*xi(
%5,3)*xi(3,3) - xi(6,3)*xi(5,5)**2 - xi(6,3))/xi(6,5);USD\\$
xi(5,4):=-(xi(6,5)*xi(5,5)*xi(5,3)
%+ xi(6,5)*xi(5,4)*xi(4,3)
+ xi(6,5)*xi(5,3)*xi(3,3) - xi(6,3)*xi(5,5)**2 - xi(6,3))/(xi(6,5)*xi(4,3))$
write "\\ USD xi(5,4):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 4x2 entry in  USD J**2 USD  gives "$
%USD  {J**2}^4_2=(xi(4,3)*xi(3,3)*xi(3,1) + xi(4,3)*xi(3,2)*xi(2,1) + xi(4,3)*xi(
%3,1)*xi(1,1) - xi(4,1)*xi(3,3)**2 - xi(4,1))/xi(2,1);USD\\$
xi(3,2):=
-(xi(4,3)*xi(3,3)*xi(3,1)
%+ xi(4,3)*xi(3,2)*xi(2,1)
+ xi(4,3)*xi(
3,1)*xi(1,1) - xi(4,1)*xi(3,3)**2 - xi(4,1))/(xi(2,1)*xi(4,3))$
write "\\ USD xi(3,2):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 6x3 entry in  USD J**2 USD  gives "$
%USD  {J**2}^6_3=xi(6,5)*xi(5,3) + xi(6,4)*xi(4,3) - xi(6,3)*xi(5,5) + xi(6,3)*xi
%(3,3);USD\\$
xi(6,4):=-(xi(6,5)*xi(5,3)
%+ xi(6,4)*xi(4,3)
- xi(6,3)*xi(5,5) + xi(6,3)*xi(3,3))/xi(4,3)$
write "\\ USD xi(6,4):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 5x1 entry in  USD J**2 USD  gives "$
%USD  {J**2}^5_1=( - xi(6,5)*xi(5,5)*xi(5,3)*xi(4,1) + xi(6,5)*xi(5,5)*xi(5,1)*xi
%(4,3) + xi(6,5)*xi(5,3)*xi(4,3)*xi(3,1) - xi(6,5)*xi(5,3)*xi(4,1)*xi(3,3) + xi(6
%,5)*xi(5,2)*xi(4,3)*xi(2,1) + xi(6,5)*xi(5,1)*xi(4,3)*xi(1,1) + xi(6,3)*xi(5,5)
%**2*xi(4,1) + xi(6,3)*xi(4,1) - xi(6,1)*xi(5,5)**2*xi(4,3) - xi(6,1)*xi(4,3))/(
%xi(6,5)*xi(4,3));USD\\$
 xi(5,2):=-
( - xi(6,5)*xi(5,5)*xi(5,3)*xi(4,1) + xi(6,5)*xi(5,5)*xi(5,1)*xi
(4,3) + xi(6,5)*xi(5,3)*xi(4,3)*xi(3,1) - xi(6,5)*xi(5,3)*xi(4,1)*xi(3,3)
%+ xi(6,5)*xi(5,2)*xi(4,3)*xi(2,1)
+ xi(6,5)*xi(5,1)*xi(4,3)*xi(1,1) + xi(6,3)*xi(5,5)
**2*xi(4,1) + xi(6,3)*xi(4,1) - xi(6,1)*xi(5,5)**2*xi(4,3) - xi(6,1)*xi(4,3))/(
xi(6,5)*xi(4,3)*xi(2,1))$
write "\\ USD xi(5,2):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%write "*****************************************************************"$
write "\\ \starline"$
write "At the present stage, we have : "$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 5x2 entry in  USD J**2 USD  gives "$
%USD  {J**2}^5_2=((xi(6,5)*xi(5,3)*xi(4,1) - xi(6,5)*xi(5,1)*xi(4,3) - xi(6,3)*xi
%(5,5)*xi(4,1) - xi(6,3)*xi(4,3)*xi(3,1) + xi(6,3)*xi(4,1)*xi(3,3) - xi(6,2)*xi(4
%,3)*xi(2,1) + xi(6,1)*xi(5,5)*xi(4,3) - xi(6,1)*xi(4,3)*xi(1,1))*(xi(5,5)**2 + 1
%))/(xi(6,5)*xi(4,3)*xi(2,1));USD\\$
xi(5,1):=
(xi(6,5)*xi(5,3)*xi(4,1)
%- xi(6,5)*xi(5,1)*xi(4,3)
- xi(6,3)*xi
(5,5)*xi(4,1) - xi(6,3)*xi(4,3)*xi(3,1) + xi(6,3)*xi(4,1)*xi(3,3) - xi(6,2)*xi(4
,3)*xi(2,1) + xi(6,1)*xi(5,5)*xi(4,3) - xi(6,1)*xi(4,3)*xi(1,1))
/(xi(6,5)*xi(4,3))$
write "\\ USD xi(5,1):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%{1,3}|6\\xi(6,5)*xi(3,3) + xi(6,5)*xi(1,1) - xi(5,5)*xi(4,3) + xi(5,5)*xi(2,1) -
% xi(4,3)*xi(1,1) + xi(3,3)*xi(2,1)\\$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ \starline \\ \4stars In the present case 1 , we suppose"$
write " USD xi(4,3) \neq xi(2,1) USD "$
write "Then, one gets from  equation USD13|6USD :"$
xi(5,5):=-(xi(3,3)*(xi(6,5)+xi(2,1))+xi(1,1)*(xi(6,5)-xi(4,3)))/(xi(2,1)-xi(4,3))$
write "\\ USD xi(5,5):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%1,4}|6\\(xi(6,5)*xi(4,3)**2*xi(2,1) - xi(6,5)*xi(4,3)*xi(2,1)**2 - xi(6,5)*xi(4
%3)*xi(1,1)**2 - xi(6,5)*xi(4,3) + xi(6,5)*xi(3,3)**2*xi(2,1) + xi(6,5)*xi(2,1)
% xi(4,3)**2*xi(1,1)**2 + xi(4,3)**2 - 2*xi(4,3)*xi(3,3)*xi(2,1)*xi(1,1) - 2*xi(
%,3)*xi(2,1) + xi(3,3)**2*xi(2,1)**2 + xi(2,1)**2)/(xi(4,3)**2 - xi(4,3)*xi(2,1)
%\\$
write "Now, equation USD{1,4}|6 USD reads USD xi(6,5) * D + N = 0 USD"$
write " where USD D:=xi(4,3)**2*xi(2,1) - xi(4,3)*xi(2,1)**2 - xi(4,3)*xi(1,1)**2 - xi(4,3)
 + xi(3,3)**2*xi(2,1) + xi(2,1) USD "$
 write "and USD N:=xi(4,3)**2*(xi(1,1)**2 + 1) - 2*xi(4,3)*xi(2,1)*(xi(3,3)*xi(1,1) +1)
+xi(2,1)**2*( xi(3,3)**2 + 1) USD"$
write "Consider USD N USD. "$
write "One has  USD N = \| Y -Z\|^2 USD"$
write "where USD Y = xi(4,3) \left( \begin{smallmatrix} xi(1,1)\\1\end{smallmatrix} \right)"$
write " \; , \;  Z = xi(2,1) \left( \begin{smallmatrix} xi(3,3)\\1\end{smallmatrix} \right)USD"$
write "Now  USD  Y \neq Z USD since  USD xi(4,3) \neq xi(2,1) USD."$
write "We conclude that  USD N \neq 0 USD. "$
write "Hence equation USD{1,4}|6 USD implies USD D  \neq 0  USD"$
write "and  USD xi(6,5):= -\frac{N}{D}  USD"$
write "Note that  USD xi(6,5)\neq 0  USD will be automatic."$
xi(6,5):=-(
  xi(4,3)**2*xi(1,1)**2 + xi(4,3)**2 - 2*xi(4,3)*xi(3,3)*xi(2,1)*xi(1,1) - 2*xi(
4,3)*xi(2,1) + xi(3,3)**2*xi(2,1)**2 + xi(2,1)**2)
/
 (xi(4,3)**2*xi(2,1) - xi(4,3)*xi(2,1)**2 - xi(4,3)*xi(1,1)**2 - xi(4,3)
 + xi(3,3)**2*xi(2,1) + xi(2,1))
$
write "\\ USD xi(6,5):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%off exp$
%on factor$
off factor$
on exp$

%WRITE "  ****Now the nonzero torsion equations left are :"$
WRITE "\\  \4stars Now the nonzero torsion equations left are :"$

COMMENT
write COLLECT_EQ$

%Latex output
 WRITE "\\ Torsion equations to cancel (Latex output) : USD"$
 for each A in COLLECT_EQ do
        if PART(A,2) neq 0 then
        <<write PART(PART(A,1),1),"|", PART(PART(A,1),2),"\\",PART(A,2),"\\">>$
 write "USD"$
write "\\ \P \\"$
%%%%%%%
%La matrice MATJ de J :
write "\par The matrix USD J USD is :\\"$

 MATRIX MATJ(DIM,DIM)$
 FOR i:=1:DIM DO FOR j:=1:DIM DO MATJ(i,j):=xi(i,j)$

 %WRITE "Matrice de J:=",MATJ$

off exp$
on factor$
 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "USD  J^",i,"_",j,"=",MATJ(i,j),";USD\\">>$

%write "matJ:=",matj$

%on exp$
%off factor$
 %%%%%%%%%%%%%%%%%
 MATJCARRE:=(MATJ)**2$

 %WRITE "Matrice de J**2:=",(MATJ)**2$


  FOR i:=1:DIM DO FOR j:=1:DIM DO
  <<WRITE "USD  {J**2}^",i,"_",j,"=",MATJCARRE(i,j),";USD\\">>$
 WRITE "\\$ det J:=",DET(MATJ)$
 WRITE "Trace J:=",TRACE(MATJ)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off exp$
on factor$
write "\\"$
write "{\fontsize{8}{10} \selectfont"$
write "\begin{equation}"$
write "\label{M10general}"$
% write "USDUSD J = \begin{pmatrix}"$
 write " J = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\end{pmatrix}" >>$
 write "\end{equation}"$
 write "}"$
 %%%%%%%%%%%%%%%%%
 write "USDUSD J^2 = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write MATJCARRE(i,j), "&">>
                         else    <<write MATJCARRE(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write MATJCARRE(i,j), "&">>
                         else    <<write MATJCARRE(i,j), "\end{pmatrix}USDUSD" >>$

 WRITE "\\USD det J:=",DET(MATJ)," USD"$
 WRITE "USD Trace J:=",TRACE(MATJ)," USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write " where USD xi(2,1)*xi(4,3) \neq 0,USD"$
write " \\ and USD D:=xi(4,3)**2*xi(2,1) - xi(4,3)*xi(2,1)**2 - xi(4,3)*xi(1,1)**2 - xi(4,3)
 + xi(3,3)**2*xi(2,1) + xi(2,1) \neq 0 USD "$

condition_D:=xi(4,3)**2*xi(2,1) - xi(4,3)*xi(2,1)**2 - xi(4,3)*xi(1,1)**2 - xi(4,3)
 + xi(3,3)**2*xi(2,1) + xi(2,1)$
write "USD D:=", condition_D, ",\neq 0 \\USD"$

write " \\ and"$
write "USD a:=xi(5,1):=", xi(5,1), ",\\USD"$
write "USD b:=xi(5,2):=", xi(5,2),",\\USD"$
write "USD c:=xi(5,4):=", xi(5,4),",\\USD"$
write "USD d:=xi(5,5):=", xi(5,5),",\\USD"$
write "USD g:=xi(5,6):=", xi(5,6),",\\USD"$
write "USD h:=xi(6,4):=", xi(6,4),",\\USD"$
write "USD k:=xi(6,5):=", xi(6,5),",\\USD"$
write "USD m:=xi(6,6):=", xi(6,6),",\\USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%% writing the specific file containing the reduce data for J %%%%%%%%%%%%%
out "matJ10_1.red"$

WRITE "% Writing the entries of J:"$
WRITE "% J(i,k) denotes the entry in the ith row , kth column"$
WRITE "% i.e. stands for  the  LaTex expression J^i_k"$
WRITE "% Similarly, xi(i,k) stands for the  LaTex expression \xi^i_k"$
 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "J(",i,",",j,"):=",MATJ(i,j)>>$


 %FOR i:=1:DIM DO FOR j:=1:DIM DO
 %<<WRITE MATJ(i,j):=MATJ(i,j)>>$

 WRITE "%where the parameters are subject to the following condition"$
 write "%USDxi(2,1)*xi(4,3) *(xi(2,1)-xi(4,3)\neq 0. USD"$
 write "%and the condition"$
 write "COMMENT%"$
write "%USD D:="$
write condition_D$
write "%,\neq 0 \\USD"$

 write "END"$
 shut "matJ10_1.red"$
 out "rprogram10_1.tex"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ check of torsion"$

 WRITE "\\Torsion equations to cancel (Latex output) : \\USD"$
 for each A in COLLECT_EQ do
        if PART(A,2) neq 0 then
        <<write PART(PART(A,1),1),"|", PART(PART(A,1),2),"\\",PART(A,2),"\\">>   $
        %<<write "\\", A,"\\">>   $
 write "USD "$

for each A in COLLECT_EQ  do if part(A,2) neq 0 then <<write part(A,1),"nonzero torsion">>$

for each A in COLLECT_EQ join if part(A,2) neq 0 then A else {}$
if length(ws) = 0 then write "zero torsion" else write "nonzero torsion"$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% off exp$
% on allfac$
on exp$
off factor$
 WRITE "\par Commutation relations of USD \mathfrak{m} : USD"$
FOR i:=1:DIM DO <<FACTOR x(i)>>$
%factor x(1),x(2),x(3),x(4),x(5),x(6)$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO X_(i,j):=x(i)*x(j)$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO Y_(i,j):=F(x(i))*F(x(j))$
FOR j:=1:DIM DO X(j):=MKID(tildex_,j)$
FOR i:=1:DIM DO <<FACTOR MKID(tildex_,i)>>$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO
    IF X_(i,j)-Y_(i,j) NEQ 0 THEN WRITE
        "\\ USD  [\tilde{x}_",i,",\tilde{x}_",j,"]=", X_(i,j) - Y_(i,j),";\\USD"$
WRITE "\P"$
FOR j:=1:DIM DO <<CLEAR X(j)>>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par Now we check if the condition USD [Jx,y]= J[x,y] USD is satisfied"$
write "USD\forall x,y \in {\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "},USD"$
write "\textit{i.e.} if USD{\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}USD"$
write "is a \textit{complex} algebra."$

for all j,k let B1(j,k) = F(x(j)*x(k))$
for all j,k let BB(j,k) = - B1(j,k) + F(x(j))*x(k)$
COLLECT_ALGEBRECOMPLEXE:=FOR j1:=1:DIM   JOIN    FOR j2:=1:DIM JOIN
   IF BB(j1,j2) NEQ 0 THEN
   {{{j1,j2},BB(j1,j2)}}
   ELSE {}$
IF LENGTH(COLLECT_ALGEBRECOMPLEXE) NEQ 0 THEN WRITE "\\USD J[x_j,x_k] \neq [Jx_j,x_k] USD
in the following cases",COLLECT_ALGEBRECOMPLEXE
   ELSE WRITE "\\USD{\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}USD is a COMPLEX ALGEBRA"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ \starline \\ \4stars "$
write "Now we consider the special case of the canonical structure:"$
xi(1,1):=0$
xi(2,1):=1$
xi(3,1):=0$
xi(4,1):=0$
xi(5,3):=0$
xi(6,1):=0$
xi(6,2):=0$
xi(6,3):=0$
xi(3,3):=0$
xi(4,3):=-1$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off exp$
on factor$

WRITE "  \\ \4stars Now the nonzero torsion equations left are :"$

COMMENT
write COLLECT_EQ$

%Latex output
 WRITE "\\ Torsion equations to cancel (Latex output) : USD"$
 for each A in COLLECT_EQ do
        if PART(A,2) neq 0 then
        <<write PART(PART(A,1),1),"|", PART(PART(A,1),2),"\\",PART(A,2),"\\">>$
 write "USD"$
write "\\ \P \\"$
%%%%%%%
%La matrice MATJ de J :
write "\par The matrix USD J USD is :\\"$

 MATRIX MATJ(DIM,DIM)$
 FOR i:=1:DIM DO FOR j:=1:DIM DO MATJ(i,j):=xi(i,j)$

 %WRITE "Matrice de J:=",MATJ$

off exp$
on factor$
 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "USD  J^",i,"_",j,"=",MATJ(i,j),";USD\\">>$

%write "matJ:=",matj$

%on exp$
%off factor$
 %%%%%%%%%%%%%%%%%
 MATJCARRE:=(MATJ)**2$

 %WRITE "Matrice de J**2:=",(MATJ)**2$


  FOR i:=1:DIM DO FOR j:=1:DIM DO
  <<WRITE "USD  {J**2}^",i,"_",j,"=",MATJCARRE(i,j),";USD\\">>$
 WRITE "\\$ det J:=",DET(MATJ)$
 WRITE "Trace J:=",TRACE(MATJ)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%
off exp$
on factor$
write "\\"$
write "{\fontsize{8}{10} \selectfont"$
write "\begin{equation}"$
% write "USDUSD J = \begin{pmatrix}"$
 write " J = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\end{pmatrix}" >>$
 write "\end{equation}"$
 write "}"$
 %%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%
 write "USDUSD J^2 = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write MATJCARRE(i,j), "&">>
                         else    <<write MATJCARRE(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write MATJCARRE(i,j), "&">>
                         else    <<write MATJCARRE(i,j), "\end{pmatrix}USDUSD" >>$

 WRITE "\\USD det J:=",DET(MATJ)," USD"$
 WRITE "USD Trace J:=",TRACE(MATJ)," USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ check of torsion"$

 WRITE "\\Torsion equations to cancel (Latex output) : \\USD"$
 for each A in COLLECT_EQ do
        if PART(A,2) neq 0 then
        <<write PART(PART(A,1),1),"|", PART(PART(A,1),2),"\\",PART(A,2),"\\">>   $
        %<<write "\\", A,"\\">>   $
 write "USD "$

for each A in COLLECT_EQ  do if part(A,2) neq 0 then <<write part(A,1),"nonzero torsion">>$

for each A in COLLECT_EQ join if part(A,2) neq 0 then A else {}$
if length(ws) = 0 then write "zero torsion" else write "nonzero torsion"$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 off exp$
 on allfac$
 WRITE "\par Commutation relations of USD \mathfrak{m} : USD"$
FOR i:=1:DIM DO <<FACTOR x(i)>>$
%factor x(1),x(2),x(3),x(4),x(5),x(6)$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO X_(i,j):=x(i)*x(j)$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO Y_(i,j):=F(x(i))*F(x(j))$
FOR j:=1:DIM DO X(j):=MKID(tildex_,j)$
FOR i:=1:DIM DO <<FACTOR MKID(tildex_,i)>>$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO
    IF X_(i,j)-Y_(i,j) NEQ 0 THEN WRITE
        "\\ USD  [\tilde{x}_",i,",\tilde{x}_",j,"]=", X_(i,j) - Y_(i,j),";\\USD"$
WRITE "\P"$
FOR j:=1:DIM DO <<CLEAR X(j)>>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par Now we check if the condition USD [Jx,y]= J[x,y] USD is satisfied"$
write "USD\forall x,y \in {\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "},USD"$
write "\textit{i.e.} if USD{\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}USD"$
write "is a \textit{complex} algebra."$

for all j,k let B1(j,k) = F(x(j)*x(k))$
for all j,k let BB(j,k) = - B1(j,k) + F(x(j))*x(k)$
COLLECT_ALGEBRECOMPLEXE:=FOR j1:=1:DIM   JOIN    FOR j2:=1:DIM JOIN
   IF BB(j1,j2) NEQ 0 THEN
   {{{j1,j2},BB(j1,j2)}}
   ELSE {}$
IF LENGTH(COLLECT_ALGEBRECOMPLEXE) NEQ 0 THEN WRITE "\\USD J[x_j,x_k] \neq [Jx_j,x_k] USD
in the following cases",COLLECT_ALGEBRECOMPLEXE
   ELSE WRITE "\\USD{\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}USD is a COMPLEX ALGEBRA"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear xi(1,1),xi(2,1),xi(3,1),xi(4,1),xi(5,3),xi(6,1),xi(6,2),xi(6,3),xi(3,3),xi(4,3)$
write "clear USDxi(1,1),xi(2,1),xi(3,1),xi(4,1),xi(5,3),xi(6,1),xi(6,2),xi(6,3),xi(3,3),xi(4,3)USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%off exp$
%on factor$
off factor$
on exp$

%WRITE "  ****Now the nonzero torsion equations left are :"$
WRITE "\\  \4stars Now the nonzero torsion equations left are :"$

COMMENT
write COLLECT_EQ$

%Latex output
 WRITE "\\ Torsion equations to cancel (Latex output) : USD"$
 for each A in COLLECT_EQ do
        if PART(A,2) neq 0 then
        <<write PART(PART(A,1),1),"|", PART(PART(A,1),2),"\\",PART(A,2),"\\">>$
 write "USD"$
write "\\ \P \\"$
%%%%%%%
%La matrice MATJ de J :
write "\par The matrix USD J USD is :\\"$

 MATRIX MATJ(DIM,DIM)$
 FOR i:=1:DIM DO FOR j:=1:DIM DO MATJ(i,j):=xi(i,j)$

 %WRITE "Matrice de J:=",MATJ$

off exp$
on factor$
 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "USD  J^",i,"_",j,"=",MATJ(i,j),";USD\\">>$

%write "matJ:=",matj$

%on exp$
%off factor$
 %%%%%%%%%%%%%%%%%
 MATJCARRE:=(MATJ)**2$

 %WRITE "Matrice de J**2:=",(MATJ)**2$


  FOR i:=1:DIM DO FOR j:=1:DIM DO
  <<WRITE "USD  {J**2}^",i,"_",j,"=",MATJCARRE(i,j),";USD\\">>$
 WRITE "\\$ det J:=",DET(MATJ)$
 WRITE "Trace J:=",TRACE(MATJ)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%off exp$
%on factor$
 write "\par Now we'll use equivalence by automorphisms."$
 write "All  automorphisms of"$
write "USD {\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}USD"$
write "are  of the following  form  :"$

matrix phi(6,6)$
for i:=1:6 do for j:=1:6 do <<phi(i,j):=b(i,j)>>$

for i:=1:4 do for j:=5:6 do <<b(i,j):=0>>$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%b(1,1):=b(1,1)$

%b(1,2):=b(1,2)$

b(1,3):=0$

b(1,4):=0$

b(1,5):=0$

b(1,6):=0$

%b(2,1):=b(2,1)$

%b(2,2):=b(2,2)$

b(2,3):=0$

b(2,4):=0$

b(2,5):=0$

b(2,6):=0$

%b(3,1):=b(3,1)$

%b(3,2):=b(3,2)$

b(3,3):=b(2,2)*b(1,1) - b(2,1)*b(1,2)$

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

b(3,5):=0$

b(3,6):=0$

%b(4,1):=b(4,1)$

%b(4,2):=b(4,2)$

b(4,3):=0$

b(4,4):=b(1,2)**2 + b(1,1)**2$

b(4,5):=0$

b(4,6):=0$

%b(5,1):=b(5,1)$

%b(5,2):=b(5,2)$

b(5,3):=b(3,2)*b(1,1) - b(3,1)*b(1,2) - b(4,1)*b(2,2) + b(4,2)*b(2,1)$

%b(5,4):=b(5,4)$

b(5,5):=(b(2,2)*b(1,1) - b(2,1)*b(1,2))*b(1,1)$

b(5,6):= - (b(2,2)*b(1,1) - b(2,1)*b(1,2))*b(1,2)$

%b(6,1):=b(6,1)$

%b(6,2):=b(6,2)$

b(6,3):= - (b(3,2)*b(2,1) - b(3,1)*b(2,2) + b(4,1)*b(1,2) - b(4,2)*b(1,1))$

%b(6,4):=b(6,4)$

b(6,5):= - (b(2,2)*b(1,1) - b(2,1)*b(1,2))*b(2,1)$

b(6,6):=(b(1,2)**2 + b(1,1)**2 + b(2,1)**2)*b(1,1) + b(2,2)*b(2,1)*b(1,2)$

%\\$ det phi:=(b(2,2)*b(1,1) - b(2,1)*b(1,2))**3*(b(2,1)**2 + b(1,1)**2)*(b(1,2)
%**2 + b(1,1)**2)**2$

%\\$  AUTOMCOND1:=(b(1,2)**2 + b(1,1)**2 + b(2,1)**2)*b(1,1) - (b(2,2)*b(1,1) - 2
%*b(2,1)*b(1,2))*b(2,2)$

%\\$  AUTOMCOND2:=(b(1,2)**2 + b(1,1)**2 - b(2,1)**2)*b(1,2) + (b(2,2)*b(1,2) + 2
%*b(2,1)*b(1,1))*b(2,2)$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

AUTOMCOND1:=(b(1,2)**2 + b(1,1)**2 + b(2,1)**2)*b(1,1) - (b(2,2)*b(1,1) - 2
*b(2,1)*b(1,2))*b(2,2)$

AUTOMCOND2:=(b(1,2)**2 + b(1,1)**2 - b(2,1)**2)*b(1,2) + (b(2,2)*b(1,2) + 2
*b(2,1)*b(1,1))*b(2,2)$
write "\\ where USD AUTOMCOND1:=",AUTOMCOND1,"USD"$
write "\\ and USD AUTOMCOND2:=",AUTOMCOND2,"USD"$
write "must vanish."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "this is equivalent to :"$
b(2,2):=-u*b(1,1)$
write "\\ USD b(2,2):=", ws,"USD"$
b(1,2):=u*b(2,1)$
write "\\ USD b(1,2):=", ws,"USD"$
u**2:=1$
write "\\ USD u**2:=", ws,"USD"$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


write "\\"$
 write "USDUSD \Phi = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write phi(i,j), "&">>
                         else    <<write phi(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write phi(i,j), "&">>
                         else    <<write phi(i,j), "\end{pmatrix}USDUSD" >>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\textit{Reduce} here checks that USD \Phi USD is indeed an automorphism:\\"$

for all i let phi_x(i) = for j:=1:6 sum phi(j,i)*x(j)$

WS:={{{1,2},phi_x(1)*phi_x(2)-phi_x(3)},
{{1,3},phi_x(1)*phi_x(3)-phi_x(5)},
{{1,4},phi_x(1)*phi_x(4)-phi_x(6)},
{{1,5},phi_x(1)*phi_x(5)},
{{1,6},phi_x(1)*phi_x(6)},
{{2,3},phi_x(2)*phi_x(3)+phi_x(6)},
{{2,4},phi_x(2)*phi_x(4)-phi_x(5)},
{{2,5},phi_x(2)*phi_x(5)},
{{2,6},phi_x(2)*phi_x(6)},
{{3,4},phi_x(3)*phi_x(4)},
{{3,5},phi_x(3)*phi_x(5)},
{{3,6},phi_x(3)*phi_x(6)},
{{4,5},phi_x(4)*phi_x(5)},
{{4,6},phi_x(4)*phi_x(6)},
{{5,6},phi_x(5)*phi_x(6)}} $

COLLECT_aut:=FOR each A in ws JOIN
   IF PART(A,2) NEQ 0 THEN {PART(A,1),PART(A,2)}
   ELSE {}$


IF LENGTH(COLLECT_aut) NEQ 0 THEN <<WRITE "USD \Phi USD is not an automorphism
 in the following cases",COLLECT_aut>>
 else <<WRITE " USD \surd \Phi USD is  an automorphism">>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ \starline \\ \4stars "$
write "In  the special case of the canonical structure:"$
xi(1,1):=0$
xi(2,1):=1$
xi(3,1):=0$
xi(4,1):=0$
xi(5,3):=0$
xi(6,1):=0$
xi(6,2):=0$
xi(6,3):=0$
xi(3,3):=0$
xi(4,3):=-1$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
matrix J2(6,6)$
r1:=phi**(-1)$
r2:=r1*MATJ$
J2:=r2*phi$
%J2:=phi**(-1)*MATJ*phi$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%on factor$
%off exp$
off factor$
on exp$
write "\\ Then USD J2:=\Phi^{-1}*J*\Phi USD has entries :"$
for i:=1:6 do for j:=1:6 do << write "\\USD J2(", i ,",", j, "):=" , J2(i,j),"USD\\">>$
write "USD det \Phi:=",det(phi),"USD"$
off factor$
on exp$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ Hence we get that the orbit of the canonical structure is 6-dimensional."$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ We return to the general case."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear xi(1,1),xi(2,1),xi(3,1),xi(4,1),xi(5,3),xi(6,1),xi(6,2),xi(6,3),xi(3,3),xi(4,3)$
write "clear USDxi(1,1),xi(2,1),xi(3,1),xi(4,1),xi(5,3),xi(6,1),xi(6,2),xi(6,3),xi(3,3),xi(4,3)USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%on factor$
%off exp$
off factor$
on exp$
write "\\ Then USD J2:=\Phi^{-1}*J*\Phi USD has the following selected entries :"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
matrix J2(6,6)$
r1:=phi**(-1)$
r2:=r1*MATJ$
%J2:=r2*phi$ heap space low occurs
J2(1,1):=for i:=1:6 sum  r2(1,i)*phi(i,1)$
write "\\ USD J2(1,1):=", ws,"USD"$
J2(3,3):=for i:=1:6 sum  r2(3,i)*phi(i,3)$
write "\\ USD J2(3,3):=", ws,"USD"$
J2(4,3):=for i:=1:6 sum  r2(4,i)*phi(i,3)$
write "\\ USD J2(4,3):=", ws,"USD"$
%J2:=phi**(-1)*MATJ*phi$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off factor$
on exp$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write " Observe first (cf ...) that USD J USD is equivalent to a structure having :"$
write "  USD xi(1,1)=0 USD and USD 0< xi(2,1)\leqslant 1 USD"$
xi(1,1):=0$
write "\\ USD xi(1,1):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ \4stars Take the following values :"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
u:=-1$
write "\\ USD u:=", ws,"USD"$
b(1,1):=1$
b(2,1):=0$
b(3,1):=0$
b(3,2):=xi(3,1)/xi(2,1)$
b(4,1):=0$
b(4,2):=xi(4,1)/xi(2,1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\USD AUTOMCOND1:=",AUTOMCOND1,"USD"$
write "\\ USD AUTOMCOND2:=",AUTOMCOND2,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


write "\\"$
 write "USDUSD \Phi = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write phi(i,j), "&">>
                         else    <<write phi(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write phi(i,j), "&">>
                         else    <<write phi(i,j), "\end{pmatrix}USDUSD" >>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\textit{Reduce} here checks that USD \Phi USD is indeed an automorphism:\\"$

for all i let phi_x(i) = for j:=1:6 sum phi(j,i)*x(j)$

WS:={{{1,2},phi_x(1)*phi_x(2)-phi_x(3)},
{{1,3},phi_x(1)*phi_x(3)-phi_x(5)},
{{1,4},phi_x(1)*phi_x(4)-phi_x(6)},
{{1,5},phi_x(1)*phi_x(5)},
{{1,6},phi_x(1)*phi_x(6)},
{{2,3},phi_x(2)*phi_x(3)+phi_x(6)},
{{2,4},phi_x(2)*phi_x(4)-phi_x(5)},
{{2,5},phi_x(2)*phi_x(5)},
{{2,6},phi_x(2)*phi_x(6)},
{{3,4},phi_x(3)*phi_x(4)},
{{3,5},phi_x(3)*phi_x(5)},
{{3,6},phi_x(3)*phi_x(6)},
{{4,5},phi_x(4)*phi_x(5)},
{{4,6},phi_x(4)*phi_x(6)},
{{5,6},phi_x(5)*phi_x(6)}} $

COLLECT_aut:=FOR each A in ws JOIN
   IF PART(A,2) NEQ 0 THEN {PART(A,1),PART(A,2)}
   ELSE {}$


IF LENGTH(COLLECT_aut) NEQ 0 THEN <<WRITE "USD \Phi USD is not an automorphism
 in the following cases",COLLECT_aut>>
 else <<WRITE " USD \surd \Phi USD is  an automorphism">>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for i:=1:6 do for j:=1:6 do << write "\\USD \Phi^", i ,"_", j, ":=" , phi(i,j),"USD\\">>$


write "\\"$
 write "USDUSD \Phi = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write phi(i,j), "&">>
                         else    <<write phi(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write phi(i,j), "&">>
                         else    <<write phi(i,j), "\end{pmatrix}USDUSD" >>$


% FOR i:=1:DIM DO FOR j:=1:DIM DO
% <<WRITE "USD  \Phi^",i,"_",j,"=",phi(i,j),";USD\\">>$

%write "phi:=",phi$

off exp$
on factor$
write "USD det \Phi:=",det(phi),"USD"$
on exp$
off factor$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
matrix J2(6,6)$
r1:=phi**(-1)$
r2:=r1*MATJ$
J2:=r2*phi$
%J2:=phi**(-1)*MATJ*phi$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%on factor$
%off exp$
off factor$
on exp$
write "\\ Then USD J2:=\Phi^{-1}*J*\Phi USD has entries :"$
for i:=1:6 do for j:=1:6 do << write "\\USD J2(", i ,",", j, "):=" , J2(i,j),"USD\\">>$
write "USD det \Phi:=",det(phi),"USD"$
off factor$
on exp$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ \4stars Hence, we are led to the case where"$
xi(3,1):=0$
write "\\ USD xi(3,1):=", ws,"USD"$
xi(4,1):=0$
write "\\ USD xi(4,1):=", ws,"USD"$
write "clear USD u, b(1,1),b(2,1),b(3,1),b(3,2),b(4,1),b(4,2) USD"$
clear  u, b(1,1),b(2,1),b(3,1),b(3,2),b(4,1),b(4,2)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for i:=1:6 do for j:=1:6 do << write "\\USD \Phi^", i ,"_", j, ":=" , phi(i,j),"USD\\">>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%
off exp$
on factor$
 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "USD  J^",i,"_",j,"=",MATJ(i,j),";USD\\">>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ \4stars Take the following values :"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
u:=-1$
write "\\ USD u:=", ws,"USD"$
b(1,1):=1$
b(2,1):=0$
b(3,1):=0$
b(3,2):=0$
b(4,1):=0$
b(4,2):=0$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\\USD J2(5,3):= - b(5,4)*xi(4,3) + xi(5,3)USD\\$
 b(5,4):=xi(5,3)/xi(4,3)$
write "\\ USD b(5,4):=", ws,"USD"$


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

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

%\\USD J2(6,3):= - b(6,4)*xi(4,3) + xi(6,3)USD\\$
b(6,4):=xi(6,3)/xi(4,3)$
write "\\ USD b(6,4):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for i:=1:6 do for j:=1:6 do << write "\\USD \Phi^", i ,"_", j, ":=" , phi(i,j),"USD\\">>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off exp$
on factor$
write "USD det \Phi:=",det(phi),"USD"$
on exp$
off factor$


r1:=phi**(-1)$
r2:=r1*MATJ$
J2:=r2*phi$
%J2:=phi**(-1)*MATJ*phi$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%on factor$
%off exp$
off factor$
on exp$
write "\\ Then USD J2:=\Phi^{-1}*J*\Phi USD has entries :"$
for i:=1:6 do for j:=1:6 do << write "\\USD J2(", i ,",", j, "):=" , J2(i,j),"USD\\">>$
write "USD det \Phi:=",det(phi),"USD"$
off factor$
on exp$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ \4stars Hence, we are led to the case where moreover "$
xi(5,3):=0$
write "\\ USD xi(5,3):=", ws,"USD"$
xi(6,1):=0$
write "\\ USD xi(6,1):=", ws,"USD"$
xi(6,2):=0$
write "\\ USD xi(6,2):=", ws,"USD"$
xi(6,3):=0$
write "\\ USD xi(6,3):=", ws,"USD"$
write "clear USD u, b(1,1),b(2,1),b(3,1),b(3,2),b(4,1),b(4,2) USD"$
write "clear USD b(5,1),b(5,2),b(5,4),b(6,4) USD"$
clear  u, b(1,1),b(2,1),b(3,1),b(3,2),b(4,1),b(4,2)$
clear  b(5,1),b(5,2),b(5,4),b(6,4) $
write "Then we have"$
for i:=5:6 do for j:=1:4 do <<write "\\ USD xi(",i,",",j,"):=",xi(i,j),"USD">>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for i:=1:6 do for j:=1:6 do << write "\\USD \Phi^", i ,"_", j, ":=" , phi(i,j),"USD\\">>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%
off exp$
on factor$
 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "USD  J^",i,"_",j,"=",MATJ(i,j),";USD\\">>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write " where USD xi(2,1)*xi(4,3) \neq 0,USD and \\"$
write "USD D:=", condition_D, ",\neq 0 .USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


write "\\"$
 write "USDUSD \Phi = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write phi(i,j), "&">>
                         else    <<write phi(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write phi(i,j), "&">>
                         else    <<write phi(i,j), "\end{pmatrix}USDUSD" >>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off exp$
on factor$
write "USD det \Phi:=",det(phi),"USD"$
on exp$
off factor$


r1:=phi**(-1)$
r2:=r1*MATJ$
J2:=r2*phi$
%J2:=phi**(-1)*MATJ*phi$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%on factor$
%off exp$
off factor$
on exp$
write "\\ Then USD J2:=\Phi^{-1}*J*\Phi USD has entries :"$
for i:=1:6 do for j:=1:6 do << write "\\USD J2(", i ,",", j, "):=" , J2(i,j),"USD\\">>$
write "USD det \Phi:=",det(phi),"USD"$
off factor$
on exp$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "From these formulae, we see that a necessary condition for equivalence"$
write "is that USD \eta^4_3=-u*xi(4,3) \; (u = \pm 1) \; and eta^3_3 =xi(3,3).USD"$
write "As  USD u=1USD would change the sign of USD xi(2,1), we conclude that"$
write "any   USD J USD with USD xi(4,3) \neq xi(2,1) is equivalent to a unique "$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%
off exp$
on factor$
write "\\"$
write "{\fontsize{8}{10} \selectfont"$
write "\begin{equation}"$
write "\label{M10final}"$
% write "USDUSD J = \begin{pmatrix}"$
 write " J(xi(2,1),xi(3,3),xi(4,3)) = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write MATJ(i,j), "&">>
                         else    <<write MATJ(i,j), "\end{pmatrix}" >>$
 write "\end{equation}"$
 write "}"$
 %%%%%%%%%%%%%%%%%
 write "USDUSD J^2 = \begin{pmatrix}"$
 for i:=1:dim do
     if i neq dim then
         for j:=1:dim do
                         if j neq dim then <<write MATJCARRE(i,j), "&">>
                         else    <<write MATJCARRE(i,j), "\\">>
     else
         for j:=1:dim do
                         if j neq dim then <<write MATJCARRE(i,j), "&">>
                         else    <<write MATJCARRE(i,j), "\end{pmatrix}USDUSD" >>$

 WRITE "\\USD det J:=",DET(MATJ)," USD"$
 WRITE "USD Trace J:=",TRACE(MATJ)," USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write " where USD 0 < xi(2,1) \leqslant 1,USD"$
write "  USD xi(2,1)*xi(4,3) \neq 0,USD"$
write " USD xi(2,1)\neq xi(4,3) ,USD"$
write "and USD D:=", condition_D, ",\neq 0 \\USD"$

write " \\ and"$
write "USD a:=xi(5,5):=", xi(5,5), ",\\USD"$
write "USD b:=xi(6,5):=", xi(6,5), ",\\USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ check of torsion"$

 WRITE "\\Torsion equations to cancel (Latex output) : \\USD"$
 for each A in COLLECT_EQ do
        if PART(A,2) neq 0 then
        <<write PART(PART(A,1),1),"|", PART(PART(A,1),2),"\\",PART(A,2),"\\">>   $
        %<<write "\\", A,"\\">>   $
 write "USD "$

for each A in COLLECT_EQ  do if part(A,2) neq 0 then <<write part(A,1),"nonzero torsion">>$

for each A in COLLECT_EQ join if part(A,2) neq 0 then A else {}$
if length(ws) = 0 then write "zero torsion" else write "nonzero torsion"$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% off exp$
% on allfac$
on exp$
off factor$
 WRITE "\par Commutation relations of USD \mathfrak{m} : USD"$
FOR i:=1:DIM DO <<FACTOR x(i)>>$
%factor x(1),x(2),x(3),x(4),x(5),x(6)$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO X_(i,j):=x(i)*x(j)$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO Y_(i,j):=F(x(i))*F(x(j))$
FOR j:=1:DIM DO X(j):=MKID(tildex_,j)$
FOR i:=1:DIM DO <<FACTOR MKID(tildex_,i)>>$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO
    IF X_(i,j)-Y_(i,j) NEQ 0 THEN WRITE
        "\\ USD  [\tilde{x}_",i,",\tilde{x}_",j,"]=", X_(i,j) - Y_(i,j),";\\USD"$
WRITE "\P"$
FOR j:=1:DIM DO <<CLEAR X(j)>>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par Now we check if the condition USD [Jx,y]= J[x,y] USD is satisfied"$
write "USD\forall x,y \in {\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "},USD"$
write "\textit{i.e.} if USD{\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}USD"$
write "is a \textit{complex} algebra."$

for all j,k let B1(j,k) = F(x(j)*x(k))$
for all j,k let BB(j,k) = - B1(j,k) + F(x(j))*x(k)$
COLLECT_ALGEBRECOMPLEXE:=FOR j1:=1:DIM   JOIN    FOR j2:=1:DIM JOIN
   IF BB(j1,j2) NEQ 0 THEN
   {{{j1,j2},BB(j1,j2)}}
   ELSE {}$
IF LENGTH(COLLECT_ALGEBRECOMPLEXE) NEQ 0 THEN WRITE "\\USD J[x_j,x_k] \neq [Jx_j,x_k] USD
in the following cases",COLLECT_ALGEBRECOMPLEXE
   ELSE WRITE "\\USD{\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}USD is a COMPLEX ALGEBRA"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ \starline"$
write "\end{document}"$
bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%