%This is "program13_45case2_2.red"   (formerly "equivstructcomplm5case22.tex") :
%computes the complex structures up to equivalence on the Lie algebra M5
%in the case 2 where B= C = 0 and in the subcase 2_2 where xi(5,5) neq 0.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%  OUTPUT   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 off echo$     off nat$
 OUT  "rprogram13_45case2_2.tex"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 ON RAT$
 OFF MSG$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%% Loading the commutation relations file  %%%%%%%%%%%%%%
DIM:= 6$
  in "6nilp/m.5"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%ECRITURE DES RELATIONS DE COMMUTATION EN TEX
WRITE "\documentclass{article}"$
WRITE "\usepackage{amsmath,amssymb}"$
WRITE "\sloppy"$
WRITE "\begin{document}"$
WRITE "This output from the file \texttt{program13_45case2_2.red}.\\"$
WRITE "Computation of all complex    structures"$
WRITE "up to equivalence on the real Lie Algebra"$
write "USD {\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}.USD"$
write "in the case 2 where B= C = 0 and in the subcase 2_2 where xi(5,5) neq 0."$
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)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 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$
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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 "USDxi(1,6):=-xi(2,5)USD"$
write "\\ and then from equation USD36|2USD : USD xi(2,6)**2 = -xi(2,5)**2USD, hence"$
write "USDxi(2,6):=0,xi(2,5):=0,xi(1,6):=0USD"$
write "But then from equation USD35|1USD : USD xi(1,5)=0USD  a contradiction"$
write "Hence USD xi(1,5)USD has to be USD 0 USD"$
xi(1,5):=0$
write "\\ USD xi(1,5):=", xi(1,5),"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):=", xi(2,5),"USD"$
write "\\ and from equation USD46|1USD :"$
xi(1,6):=0$
write "\\ USD xi(1,6):=", xi(1,6),"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):=", xi(2,6),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Suppose now USDxi(4,6) \neq 0USD."$
write "\\ Then one first gets"$
write "\\ from equation USD26|4USD : which reads USD (xi(4,5)-xi(3,6))xi(4,6)=0USD"$
write "USDxi(4,5):=xi(3,6)USD"$
write "\\ and then from equation USD16|4USD"$
write "which reads USD xi(4,6)$**2 + xi(4,5)xi(3,6)=0USD"$
write ": USD xi(4,6)**2 = -xi(3,6)**2USD, hence"$
write "USDxi(4,6):=0,xi(3,6):=0USD"$
write "  a contradiction"$
write "Hence USD xi(4,6)USD has to be USD 0 USD"$
xi(4,6):=0$
write "\\ USD xi(4,6):=", xi(4,6),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "then, one gets from  equation USD25|4USD :"$
xi(4,5):=0$
write "\\ USD xi(4,5):=", xi(4,5),"USD"$
write "\\ and from equation USD26|6USD :"$
xi(3,6):=0$
write "\\ USD xi(3,6):=", xi(3,6),"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):=", xi(3,5),"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 "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 6x1 entry in  USD J**2 USD  gives "$

%USD  J**2^6_1=xi(6,5)*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);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)$
write "\\ USD xi(5,1):=", ws,"USD"$

write "and the 6x2 entry in  USD J**2 USD  gives "$
%USD  J**2^6_2=xi(6,5)*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);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)$
write "\\ USD xi(5,2):=", ws,"USD"$

write "and 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) + xi(6,2)*xi(2,3) + xi(6,1)*xi(1,3);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)$
write "\\ USD xi(5,3):=", ws,"USD"$

write "and the 6x4 entry in  USD J**2 USD  gives "$
%USD  J**2^6_4=xi(6,5)*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);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)$
write "\\ USD xi(5,4):=", ws,"USD"$
write "From the trace of USD JUSD, we get:"$
xi(4,4):=-for i:=1:3 sum xi(i,i)$
write "\\ USD xi(4,4):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "*****************************************************************"$
write "\starline"$
write "At the present stage, we have : "$
off exp$
on factor$

WRITE " \par **** \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)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "*****************************************************************"$
write "\starline"$
write "Note that USD xi(5,j) USD is a linear combination of the USD xi(6,j)USD's"$
write "USD 1\leq j \leq 4 USD"$
write "hence vanish if all USD xi(6,j)USD's do"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%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>>$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %this computation of automorphism from  rautomtotal4.red

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

b(1,2):=u*b(2,1)$

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

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

b(1,5):=0$

b(1,6):=0$


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


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

b(2,5):=0$

b(2,6):=0$

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

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

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

b(3,4):=u*b(4,3)$

b(3,5):=0$

b(3,6):=0$


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


b(4,4):=-u*b(3,3)$

b(4,5):=0$

b(4,6):=0$

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

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

%b(5,3):=b(5,3)$

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

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

b(5,6):=-(b(3,3)*b(2,1) - b(3,1)*b(2,3) + b(4,1)*b(1,3) - b(4,3)*b(1,1))*u$

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

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

%b(6,3):=b(6,3)$

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

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

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


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 "where USDu**2=1 USD\\"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "****Take the following values :"$
%deferred after the latex writing down of Phi for printing reason
write "USD b(5,j):="$
write "\frac {-xi(6,j)+ \sum_{k=1}^{4} b(6,k)*xi(k,j) -xi(6,6)*b(6,j)}{xi(6,5)} USD"$
write "and :"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
u:=-1$
b(1,1):=1$
b(1,3):=0$
b(2,1):=0$
b(2,3):=0$
b(3,1):=0$
b(4,1):=0$
b(3,3):=1$
b(4,3):=0$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


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$

%for j:=1:4 do
%<<b(5,j):=(-xi(6,j)+ (for k:=1:4 sum b(6,k)*xi(k,j)) -xi(6,6)*b(6,j))/xi(6,5)>>$
%write "USD b(5,j):="$

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$
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 "Note that in USD J2 USD the 5xj and 6xj terms vanish "$
write "(USD1 \leqslant j \leqslant 4)USD from the remaining equations as it must be"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "**** \4stars Hence, we are led to the case where"$
write "USD xi(5,j)=xi(6,j) = 0 \forall j \, 1 \leqslant j \leqslant 4 USD"$
for i:=5:6 do for j:=1:4 do  <<xi(i,j):=0>>$
clear 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$
write "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"$
for i:=5:6 do for j:=1:4 do <<write "xi(",i,",",j,"):=",xi(i,j)>>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "*****************************************************************"$
write "\starline"$
write "**** Until this stage, there was no difference with case 1"$
write "*****************************************************************"$
write "\starline"$
write "**** In the present  case 2"$
write "We now suppose USD B=C=0 USD"$
xi(1,3):=0$
write "\\ USD xi(1,3):=", ws,"USD"$
xi(1,4):=0$
write "\\ USD xi(1,4):=", ws,"USD"$
xi(2,3):=0$
write "\\ USD xi(2,3):=", ws,"USD"$
xi(2,4):=0$
write "\\ USD xi(2,4):=", ws,"USD"$
xi(3,1):=0$
write "\\ USD xi(3,1):=", ws,"USD"$
xi(3,2):=0$
write "\\ USD xi(3,2):=", ws,"USD"$
xi(4,1):=0$
write "\\ USD xi(4,1):=", ws,"USD"$
xi(4,2):=0$
write "\\ USD xi(4,2):=", ws,"USD"$

%for i:=5:6 do for j:=1:4 do <<xi(i,j):=0>>$
%for i:=5:6 do for j:=1:4 do <<write "xi(",i,",",j,"):=",xi(i,j)>>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Then the 1x1 entry in  USD J**2 USD  gives that xi(1,2)xi(2,1) and \neq 0"$
xi(1,2):= (-1-xi(1,1)**2)/xi(2,1)$
write "\\ USD xi(1,2):=", ws,"USD"$
%USD  {J**2}^1_1=xi(2,1)*xi(1,2) + xi(1,1)**2;USD\\$
write "The 1x2 entry in  USD J**2 USD  gives"$
%USD  {J**2}^1_2=xi(1,2)*(xi(2,2) + xi(1,1));USD\\$
xi(2,2):= -xi(1,1)$
write "\\ USD xi(2,2):=", ws,"USD"$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "**** Then the 3x3 entry in  USD J**2 USD  gives that xi(3,4)xi(4,3) and \neq 0"$
xi(3,4):= (-1-xi(3,3)**2)/xi(4,3)$
write "\\ USD xi(3,4):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

write "*****************************************************************"$
write "\starline"$
write "Now the matrix  USD J USD is"$
 %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\\">>$
on exp$
off factor$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "*****************************************************************"$
write "\starline"$
write "By equivalence, we can suppose USD xi(1,1):=0 , xi(3,3):=0 USD"$
xi(1,1):=0$
write "\\ USD xi(1,1):=", ws,"USD"$
xi(3,3):=0$
write "\\ USD xi(3,3):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "**** \4stars In the present"$
write "case 2.2"$
write "Suppose now USDxi(5,5) \neq 0USD."$
write "\\ Then one first gets"$
write "\\ from equation USD13|6USD : which reads USD (xi(4,3)-xi(2,1))xi(5,5)=0USD"$
write "USDxi(4,3):=xi(2,1)USD"$
xi(4,3):=xi(2,1)$
write "\\ USD xi(4,3):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "\\ Then equation USD13|5USD  is USD -xi(2,1)**2+1=0USD"$
xi(2,1)**2:=1$
write "\\ USD xi(2,1)**2:=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off exp$
on factor$
WRITE "\par 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$
 %%%%%%%%%%%%%%%%%
write "\\"$
%write "{fontsize{8}{10} \selectfont"$
 write "USDUSD 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}USDUSD" >>$
%                         write "}"$
 %%%%%%%%%%%%%%%%%
 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)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%
 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 "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"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
on exp$
off factor$
% 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)>>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
on exp$
off factor$
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 "\end{document}"$
bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%