off echo,nat$
out "rcalculderivgtildedelta6_nxC3case5IV.r"$
write "rcalculderivgtildedelta6_nxC3case5IV.r"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%copy here the file reducparautommodg***.red
% from  : The generic derivation as in (Cohomology tables section ******) :
% until : the instruction write "paramindexeslist:=",paramindexeslist$
%and delete the  3 lines pertaining to phi just before that last instruction.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generic derivation as computed by geneLplus.tex :
operator xi$

delta:=
mat((xi(1,1),xi(1,2),0,0,0,0),
(xi(2,1),xi(2,2),0,0,0,0),
(xi(3,1),xi(3,2),xi(1,1)+xi(2,2),xi(3,4),xi(3,5),xi(3,6)),
(xi(4,1),xi(4,2),0,xi(4,4),xi(4,5),xi(4,6)),
(xi(5,1),xi(5,2),0,xi(5,4),xi(5,5),xi(5,6)),
(xi(6,1),xi(6,2),0,xi(6,4),xi(6,5),xi(6,6)))$

write "generic derivation : delta:=",delta;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The nonzero adjoint derivations
matrix adx1(6,6)$
adx1:=
sub({
xi(1,1)=0,xi(1,2)=0,
xi(2,1)=0,xi(2,2)=0,
xi(3,1)=0,xi(3,2)=1,xi(3,4)=0,xi(3,5)=0,xi(3,6)=0,
xi(4,1)=0,xi(4,2)=0,xi(4,4)=0,xi(4,5)=0,xi(4,6)=0,
xi(5,1)=0,xi(5,2)=0,xi(5,4)=0,xi(5,5)=0,xi(5,6)=0,
xi(6,1)=0,xi(6,2)=0,xi(6,4)=0,xi(6,5)=0,xi(6,6)=0}, delta)$
matrix adx2(6,6)$
adx2:=
sub({
xi(1,1)=0,xi(1,2)=0,
xi(2,1)=0,xi(2,2)=0,
xi(3,1)=-1,xi(3,2)=0,xi(3,4)=0,xi(3,5)=0,xi(3,6)=0,
xi(4,1)=0,xi(4,2)=0,xi(4,4)=0,xi(4,5)=0,xi(4,6)=0,
xi(5,1)=0,xi(5,2)=0,xi(5,4)=0,xi(5,5)=0,xi(5,6)=0,
xi(6,1)=0,xi(6,2)=0,xi(6,4)=0,xi(6,5)=0,xi(6,6)=0}, delta)$
%matrix adx3(6,6)$
%adx3:=
%matrix adx4(6,6)$
%adx4:=
%matrix adx5(6,6)$
%adx5:=
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
on nat$
write adx1:=adx1$
write adx2:=adx2$
%write adx3:=adx3$
%write adx4:=adx4$
%write adx5:=adx5$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "The generic nilpotent derivation : "$
%write "The generic nilpotent derivation : the eigenvalues are 0"$
write "The matrices MATA:=((xi(1,1),xi(1,2)),(xi(2,1),xi(2,2))) is nilpotent"$
MATA:=mat((xi(1,1),xi(1,2)),(xi(2,1),xi(2,2)))$

write "And  MATB:=((xi(4,4),xi(4,5),xi(4,6)),(xi(5,4),xi(5,5),xi(5,6)),(xi(6,4),xi(6,5),xi(6,6))) are nilpotent"$
MATB:=mat((xi(4,4),xi(4,5),xi(4,6)),(xi(5,4),xi(5,5),xi(5,6)),(xi(6,4),xi(6,5),xi(6,6)))$
MATC:=mat((xi(3,4)),(xi(3,5)),(xi(3,6)))$
MATD:=mat((xi(4,1),xi(4,2)),(xi(5,1),xi(5,2)),(xi(6,1),xi(6,2)))$
write "(WE denote MATA,MATB, since B is already for the entries of phi)"$
write "***We consider here the case 5 where MATA has nilpotent order 1"$
write " and MATB has nilpotent order 1.***"$
%write "In that case, one may suppose MATA:=((0,1),(0,0))."$
xi(1,1):=0$
xi(1,2):=1$
xi(2,1):=0$
xi(2,2):=0$
for i:=1:2 do for j:=1:2 do <<write "xi(",i,",",j,"):= ",xi(i,j)>>$
for i:=4:6 do for j:=4:6 do xi(i,j):=0$
clear xi(4,5)$
xi(4,5):=1$
%clear xi(5,6)$
%xi(5,6):=1$
write "MATB:= ",MATB;
write "MATC:= ",MATC;
write "MATD:= ",MATD;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% by subtracting adjoints one then may suppose
xi(3,1):=0$
xi(3,2):=0$
write "by subtracting adjoints one then may suppose:"$
write "xi(3,1):=0,xi(3,2):=0."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "delta:=",delta;
write "We denote this delta by the shortform"$
shortformdelta:=
{xi(4,1),xi(4,2),SS,xi(5,1),xi(5,2),SS,xi(6,1),xi(6,2),SS,xi(3,4),xi(3,5),xi(3,6) }$
paramindexeslist:=
{{4,1},{4,2},{5,1},{5,2},{6,1},{6,2},{3,4},{3,5},{3,6}}$

write "shortformdelta:=", shortformdelta$
write "paramindexeslist:=",paramindexeslist$

off nat$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%initial conditions for specific delta

xi(4,1):=a$
xi(4,2):=0$

xi(5,1):=0$
xi(5,2):=0$

xi(6,1):=0$
xi(6,2):=0$

xi(3,4):=0$
xi(3,5):=0$
xi(3,6):=1$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
a:=0$
%b:=0$

%conditionssura:={}$
%conditionssurb:={}$
%write "a neq ",conditionssura$
%write "b neq ",conditionssurb$
write "a:=",a$
%write "b:=",b$
on nat$
write "delta:=",delta$
off nat$
write "shortformdelta:=",shortformdelta$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% in "the commut relations file of g"
% extended to the commut. relations of $\tilde{g}_delta
%by the commutation relations involving X(0)=delta
% and the projector V(0) on X(0)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

operator x,v$
in "6nilp/6nilp.nxC3"$    %%%%%%%%% DON T FORGET TO ENTER THE SUITABLE FILE!!!!!

for j:=0:6 do x(0)*x(j):= if j neq 0 then  for k:=1:6 sum delta(k,j)*x(k) else 0
for j:=1:6 do x(j)*x(0):= - x(0)*x(j)$

FOR j:=0:DIM DO
       V(0)*x(j):= IF j=0 THEN 1 ELSE 0;
FOR i:=1:DIM DO  V(i)*x(0):=0$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%MATRIXD will denote the generic derivation of the 7-dimensional {\tilde{g}}_{delta}
% Let MATRIXD=matrix(D(i,j)). Collecting the equations {i,j}   1\leqslant i < i \leqslant 6
%              -D [X(i),X(j)] + [DX(i), X(j)] + [X(i), DX(j)].
MATRIX MATRIXD(7,7)$
operator d$
for i:=1:7 DO for j:=1:7  DO MATRIXD(i,j):=D(i-1,j-1)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
COLLECT_DERIVGTILDEDELTA:=
for i:=0:DIM-1 JOIN for j:=i+1:DIM JOIN
     { {{i,j},
            -(WS:=for k:=0:DIM sum
            (WS:=V(k)*(WS:=X(i)*X(j))) * (for m:=0:DIM sum D(m,k)*X(m))
            )
            + (WS:=for k:=0:DIM sum D(k,i)*(WS:=X(k)*X(j)))
            + (WS:= for k:=0:DIM sum D(k,j)*(WS:=X(i)*X(k)))
            }}$
%COLLECT_DERIVGTILDEDELTA;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Collecting the nonzero equations {i,j}|k obtained by projecting
%on X(k) the derivation equation -D [X(i),X(j)] + [DX(i), X(j)] + [X(i), DX(j)].
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%off exp$
%on factor$

%la liste des équations non nulles
%write "list of the nonzero derivation equations"$

COLLECT_EQ:=FOR j1:=1:LENGTH(COLLECT_DERIVGTILDEDELTA) JOIN
   FOR j2:=0:DIM JOIN
   IF V(j2)*PART(PART(COLLECT_DERIVGTILDEDELTA,j1),2) NEQ 0 THEN
         {{{PART(PART(COLLECT_DERIVGTILDEDELTA,j1),1),j2},
                  V(j2)*PART(PART(COLLECT_DERIVGTILDEDELTA,j1),2)} }
   ELSE  {}$

COMMENT
 WRITE "Derivation 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$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE OTEZERO(U)$  %enleve les 0 dans une liste
BEGIN$
LIST U$
RETURN FOR EACH A IN U JOIN IF A NEQ 0 THEN {A} ELSE {}$
END$


PROCEDURE distinct(U)$
BEGIN;INTEGER j;LIST UU$ j:=1;UU:={}$
S: IF U NEQ {} THEN    ZZZ:=PART(u,j)$
IF ZZZ MEMBER UU THEN <<GO TO P>> ELSE UU:= ZZZ. UU $
P: IF j<LENGTH(U) THEN <<j:=j+1,GO TO S>> ELSE <<RETURN REVERSE(UU)>>$
CLEAR ZZZ$
END$

PROCEDURE UNKNOWNSINLIST(U)$
BEGIN$ LIST U$
RETURN  IF U={} THEN U ELSE
        IF PART(PART(SOLVE(U),1),0) = LIST
          THEN FOR EACH A IN PART(SOLVE(U),1) COLLECT LHS A
        ELSE {LHS PART(SOLVE(U),1)} $
END$


PROCEDURE UNKNOWNSINEXPRESSION(U)$
BEGIN$
RETURN  IF NUMBERP U THEN {}  ELSE
        IF PART(PART(SOLVE(U),1),0) = LIST
          THEN FOR EACH A IN PART(SOLVE(U),1) COLLECT LHS A
        ELSE {LHS PART(SOLVE(U),1)} $
END$


PROCEDURE SSOLVE(n,p)$
BEGIN INTEGER k,j$
 k:=1$ j:=1$
S1: IF z(k,j) NEQ 0 THEN <<GO TO S2>> ELSE <<GO TO S3>>$
S2: CALLLET(z(k,j),0) $
S3: IF j<p THEN j:=j+1 ELSE IF k<n  THEN <<k:=k+1$ j:=1>> ELSE GO TO
 S4$
GO TO S1$
S4: END$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE SELECTVAR(U)$
BEGIN$
integer i,j$
i:=6$ j:=6$
LI:=UNKNOWNSINEXPRESSION(U)$
S: IF D(i,j) MEMBER LI AND NUMBERP DF(U,D(i,j))
   THEN <<RETURN D(i,j), GO TO F>>
   ELSE
        IF j>0 THEN <<j:=j-1, GO TO S>>
        ELSE IF i>0 THEN <<i:=i-1,j:=6, GO TO S>>
             ELSE <<write "pas de selection possible de variable a coefficient numerique dans ", U, return 0>>$
F :
END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
procedure calllet(U,V)$
let U=V$


PROCEDURE  resol(L)$
begin  integer k$
k:=1$
S:
  AA:=part(part(L,k),2)$
  If AA=0 then <<go to F>>$
  write "on resout l'equation " , PART(PART(L,k),1)," qui est  maintenant AA:=",AA$
  W:= SELECTVAR(AA)$
  if W=0 then <<go to F>>$
  write  "bonne inconnue W:=",W$
  WW:= RHS(part(solve(AA=0,W),1))$
  write "sa valeur doit etre WW:=",WW$
  calllet(W,WW)$
  F:
if k < length(L) then <<k:=k+1, go to S>>$
     clear AA,W,WW$
END$


resol(COLLECT_EQ)$



off factor$
 WRITE "Derivation equations to cancel (Reduce output) : \\", COLLECT_EQ$
% bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%write "Il y a une phase 2"$
write "Il n'y a pas de phase 2"$
%write "a neq ",conditionssura$
%write "b neq ",conditionssurb$

%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


matrix MATD(7,7)$
for i:=1:7 do for j:=1:7 do MATD(i,j):=D(i-1,j-1)$

off  nat$
write "derivation generique de gtildedelta:"$
write "MATD:=",MATD$
on nat$
write "pour delta:=",delta$
write "pour shortformdelta:=",shortformdelta$
off nat$



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
listMATD:=for i:=1:7 join for j:=1:7 collect MATD(i,j)$
listparametresMATDincluantparamalgebre_a:=UNKNOWNSINLIST(listmatD)$
listparametresMATD:=
FOR EACH U IN listparametresMATDincluantparamalgebre_a
               join if U neq a and U neq b then {U} ELSE {}$
write "listeparametresMATD",WS$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
operator dd$
LISTparametrestemoinMATD:=
for each h in listparametresMATD collect dd(part(h,1),part(h,2))$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "dim Der(gtildedelta):=",LENGTH(listparametresMATD)$
%bye$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%  here are the procedures we use to compute the maximal torus %%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%   ecriture du premier membre du tore
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%ecrit le tore t1=D(i,j)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE CALCUL_t1(i,j)$ %fait d(i,j)=1 autres nuls dans la matrice MATD
BEGIN$
integer i,j$

%WS:=for each U in listparametresMATD  JOIN IF U neq d(i,j) and U neq a then {U} else {}$
%for each U in WS DO <<calllet(U,0)>>$


WS:= listparametresMATD$
WS:= for each U in WS JOIN IF U neq d(i,j) and arglength(U)= 2 then {U} else {}$
for each U in WS DO <<calllet(U,0)>>$

d(i,j):=1$
off nat$
WS:=MATD;

for each U in listparametrestemoinMATD do <<clear d(part(U,1),part(U,2))>>$
on nat$
write "t1:=D(",i,",",j,"):=",ws$
off nat$
rkgtildedelta:=1$
RETURN WS$

END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE COMMUTANT(t1)$
BEGIN$
Z:=MATD*t1-t1*MATD$
SSOLVE(7,7)$
WS1:=MATD$

listMATD:=for i:=1:7 join for j:=1:7 collect MATD(i,j)$

listparametresCOMMUTANT_t1_incluantparamalgebre_a:=UNKNOWNSINLIST(listmatD)$
listparametresCOMMUTANT_t1:=
FOR EACH U IN listparametresCOMMUTANT_t1_incluantparamalgebre_a join if U neq a then {U} ELSE {}$

listparametrestemoinsresolusdanscalculCOMMUTANT_t1 :=
             for each h in LISTparametrestemoinMATD
                join IF d(part(h,1),part(h,2)) MEMBER listparametresCOMMUTANT_t1
                     THEN {}
                     ELSE {dd(part(h,1),part(h,2))} $


for each U in listparametrestemoinsresolusdanscalculCOMMUTANT_t1 do
<<clear d(part(U,1),part(U,2))>>$

write "commutant de t1 dans der(gtildedelta):"$
%write "mateigent du commutant de t1 dans der(gtildedelta):",mateigen(ws1,x)$
RETURN WS1$
END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%   ecriture du second membre du tore
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE CALCUL_t2(i,j)$
%fait d(i,j)=1 autres nuls dans la matrice restante MATD dans le commutant de t1
BEGIN$
integer i,j$
% calcule d'abord le matrices MATD du commutant de t1
Z:=MATD*t1-t1*MATD$
SSOLVE(7,7)$
WS1:=UNKNOWNSINLIST(listmatD)$
%ws:= FOR EACH U IN ws join if U neq a and U neq b then {U} ELSE {}$
IF d(i,j) member ws1 then
                     else
           <<write "d(",i,",",j,")  not an option for calcul_t2(i,j) ************"$
                 rederr "********** invalid argument for  calcul_t2(i,j) ************">>$
WS:= for each U in WS1 JOIN IF U neq d(i,j) and arglength(U)= 2 then {U} else {}$
for each U in WS DO <<calllet(U,0)>>$

d(i,j):=1$
off nat$
WS:=MATD;

for each U in listparametrestemoinMATD do <<clear d(part(U,1),part(U,2))>>$
%restaure les parametres de MATD
on nat$
write "t2:=D(",i,",",j,"):=",ws$
off nat$
rkgtildedelta:=2$
RETURN WS$

END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE COMMUTANT2(t1,t2)$
BEGIN$
Z:=MATD*t1-t1*MATD$
SSOLVE(7,7)$
Z:=MATD*t2-t2*MATD$
SSOLVE(7,7)$
WS1:=MATD$

listMATD:=for i:=1:7 join for j:=1:7 collect MATD(i,j)$

listparametresCOMMUTANT_t1_t2_incluantparamalgebre_a:=UNKNOWNSINLIST(listmatD)$
listparametresCOMMUTANT_t1_t2:=
FOR EACH U IN listparametresCOMMUTANT_t1_t2_incluantparamalgebre_a join if U neq a then {U} ELSE {}$

listparametrestemoinsresolusdanscalculCOMMUTANT_t1_t2 :=
             for each h in LISTparametrestemoinMATD
                join IF d(part(h,1),part(h,2)) MEMBER listparametresCOMMUTANT_t1_t2
                     THEN {}
                     ELSE {dd(part(h,1),part(h,2))} $


for each U in listparametrestemoinsresolusdanscalculCOMMUTANT_t1_t2 do
<<clear d(part(U,1),part(U,2))>>$

write "commutant simultane de t1,t2 dans der(gtildedelta):"$
RETURN WS1$
END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%   ecriture du troiseme membre du tore
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE COMMUTANT3(t1,t2,t3)$
BEGIN$
Z:=MATD*t1-t1*MATD$
SSOLVE(7,7)$
Z:=MATD*t2-t2*MATD$
SSOLVE(7,7)$
Z:=MATD*t3-t3*MATD$
SSOLVE(7,7)$
WS1:=MATD$

listMATD:=for i:=1:7 join for j:=1:7 collect MATD(i,j)$

listparametresCOMMUTANT_t1_t2_t3_incluantparamalgebre_a:=UNKNOWNSINLIST(listmatD)$
listparametresCOMMUTANT_t1_t2_t3:=
FOR EACH U IN listparametresCOMMUTANT_t1_t2_t3_incluantparamalgebre_a join if U neq a then {U} ELSE {}$

listparametrestemoinsresolusdanscalculCOMMUTANT_t1_t2_t3 :=
             for each h in LISTparametrestemoinMATD
                join IF d(part(h,1),part(h,2)) MEMBER listparametresCOMMUTANT_t1_t2_t3
                     THEN {}
                     ELSE {dd(part(h,1),part(h,2))} $


for each U in listparametrestemoinsresolusdanscalculCOMMUTANT_t1_t2_t3 do
<<clear d(part(U,1),part(U,2))>>$

write "commutant simultane de t1,t2,t3 dans der(gtildedelta):"$
RETURN WS1$
END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE CALCUL_t3(i,j)$
%fait d(i,j)=1 autres nuls dans la matrice restante MATD dans le commutant de t1 et t2
BEGIN$
integer i,j$
% calcule d'abord le matrices MATD du commutant de t1 et t2
Z:=MATD*t1-t1*MATD$
SSOLVE(7,7)$
Z:=MATD*t2-t2*MATD$
SSOLVE(7,7)$


WS1:=UNKNOWNSINLIST(listmatD)$
%ws:= FOR EACH U IN ws join if U neq a and U neq b then {U} ELSE {}$
IF d(i,j) member ws1 then
                     else
           <<write "d(",i,",",j,")  not an option for calcul_t3(i,j) ************"$
                 rederr "********** invalid argument for  calcul_t3(i,j) ************">>$
WS:= for each U in WS1 JOIN IF U neq d(i,j) and arglength(U)= 2 then {U} else {}$
for each U in WS DO <<calllet(U,0)>>$

d(i,j):=1$

off nat$
WS:=MATD;

for each U in listparametrestemoinMATD do <<clear d(part(U,1),part(U,2))>>$
%restaure les parametres de MATD
on nat$
write "t3:=D(",i,",",j,"):=",ws$
off nat$
rkgtildedelta:=3$
RETURN WS$

END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE COMMUTANT4(t1,t2,t3,t4)$
BEGIN$
Z:=MATD*t1-t1*MATD$
SSOLVE(7,7)$
Z:=MATD*t2-t2*MATD$
SSOLVE(7,7)$
Z:=MATD*t3-t3*MATD$
SSOLVE(7,7)$
Z:=MATD*t4-t4*MATD$
SSOLVE(7,7)$
WS1:=MATD$

listMATD:=for i:=1:7 join for j:=1:7 collect MATD(i,j)$

listparametresCOMMUTANT_t1_t2_t3_t4incluantparamalgebre_a:=UNKNOWNSINLIST(listmatD)$

listparametresCOMMUTANT_t1_t2_t3_t4:=
FOR EACH U IN listparametresCOMMUTANT_t1_t2_t3_t4incluantparamalgebre_a join if U neq a then {U} ELSE {}$

listparametrestemoinsresolusdanscalculCOMMUTANT_t1_t2_t3_t4 :=
             for each h in LISTparametrestemoinMATD
                join IF d(part(h,1),part(h,2)) MEMBER listparametresCOMMUTANT_t1_t2_t3_t4
                     THEN {}
                     ELSE {dd(part(h,1),part(h,2))} $


for each U in listparametrestemoinsresolusdanscalculCOMMUTANT_t1_t2_t3_t4 do
<<clear d(part(U,1),part(U,2))>>$

write "commutant simultane de t1,t2,t3,t4 dans der(gtildedelta):"$
RETURN WS1$
END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE CALCUL_t4(i,j)$
%fait d(i,j)=1 autres nuls dans la matrice restante MATD dans le commutant de t1 et t2
BEGIN$
integer i,j$
% calcule d'abord le matrices MATD du commutant de t1 et t2 et t3
Z:=MATD*t1-t1*MATD$
SSOLVE(7,7)$
Z:=MATD*t2-t2*MATD$
SSOLVE(7,7)$
Z:=MATD*t3-t3*MATD$
SSOLVE(7,7)$
write "MATD:=",MATD$
WS1:=UNKNOWNSINLIST(listmatD)$


IF d(i,j) member UNKNOWNSINLIST(listmatD) then else
<<write "d(",i,",",j,")  not an option for calcul_t4(i,j) ************"$
rederr "********** invalid argument for  calcul_t4(i,j) ************">>$

WS:= for each U in WS1 JOIN IF U neq d(i,j) and arglength(U)= 2 then {U} else {}$
for each U in WS DO <<calllet(U,0)>>$

d(i,j):=1$

off nat$
WS:=MATD;

for each U in listparametrestemoinMATD do <<clear d(part(U,1),part(U,2))>>$
%restaure les parametres de MATD
on nat$
write "t4:=D(",i,",",j,"):=",ws$
off nat$
rkgtildedelta:=4$
RETURN WS$

END$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%introduction de la matrice ID identite
matrix ID(7,7)$
for i:=1:7 do for j:=1:7 do <<ID(i,j):=if i=j then 1 else 0>>$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%introduction de la matrice nullmatrix nulle
matrix NULLMATRIX(7,7)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PROCEDURE CARACT_NILP_ALG_TEST()$
%this procedure makes the test for caracteristically nilpotent Lie algebra
BEGIN$
integer j$
j:=2$
for j:=1:7 do <<write "MATD**",j,":=",MATD**(j)>>$

S :
WS:=MATD**(j)$
<<write "MATD**",j,":=",WS>>$
IF WS = NULLMATRIX THEN
  <<write "***********   gtildedelta est caracteristiquement nilpotente d'ordre",j>>
        ELSE
  <<j:=j+1$ IF j LEQ 7 THEN <<GO TO S>> ELSE
  <<write "***********   gtildedelta n'est pas caracteristiquement nilpotente">> >>$
END$
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%%%%%%%%   TEST for caracteristically nilpotent Lie algebra
% CARACT_NILP_ALG_TEST()$
% bye$

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%%%%%%%%   ecriture du premier membre du tore
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%write "un element t1 d'un tore "$

t1:=calcul_t1(0,0)$

on nat$

%%
%off nat$
%on factor$
%det(t1-x*ID);
%off factor$
%mateigen(t1,x);


%write COMMUTANT(t1)$
%bye$
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%%%%%%%%   ecriture du second membre du tore
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t2:=calcul_t2(1,1)$

%%off nat$
on factor$
%det(t2-x*ID);
%off factor$
on nat$
%mateigen(t2,x);
off nat$


%write COMMUTANT2(t1,t2)$
on nat$
%bye$
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%%%%%%%%   ecriture du troiseme membre du tore
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t3:=calcul_t3(4,4)$

%%off nat$
%on factor$
%det(t3-x*ID);
%off factor$
on nat$
%mateigen(t3,x);

write COMMUTANT3(t1,t2,t3)$
%bye$

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%t4:=calcul_t4(0,1)$
on nat$

%%off nat$
%on factor$
%det(t4-x*ID);
%off factor$
%mateigen(t4,x);
%bye$

%write COMMUTANT4(t1,t2,t3,t4)$
%bye$




IF rkgtildedelta=1
   THEN write "rank 1 with maximal torus t1"
ELSE IF rkgtildedelta=2 THEN write "rank 2 with maximal torus t1,t2"
ELSE IF rkgtildedelta=3 THEN write "rank 3 with maximal torus t1,t2,t3"
ELSE IF rkgtildedelta=4 THEN write "rank 4 with maximal torus t1,t2,t3,t4"
ELSE IF rkgtildedelta=4 THEN write "rank 4 with maximal torus t1,t2,t3,t4"
ELSE write "rank 0 :gtildedelta is characteristically nilpotent"$
write rkgtildedelta$

write "matrice de passage de la base X(0)=delta, X(1),..., X(6) a une base  diagonalisant le tore maximal: on peut prendre"$


matrix P(7,7)$
P:=ID$
%on complex$
%for i:=1:7 do <<P(i,i):=1>>$

on nat$

write "P:=",P$
IF rkgtildedelta=1
   THEN write "P**(-1)*t1*P:=", P**(-1)*t1*P
ELSE IF rkgtildedelta=2
   THEN write "P**(-1)*t1*P:=", P**(-1)*t1*P,"P**(-1)*t2*P:=", P**(-1)*t2*P
ELSE IF rkgtildedelta=3
   THEN write "P**(-1)*t1*P:=", P**(-1)*t1*P,"P**(-1)*t2*P:=", P**(-1)*t2*P
              ,"P**(-1)*t3*P:=", P**(-1)*t3*P
ELSE IF rkgtildedelta=4
   THEN write "P**(-1)*t1*P:=", P**(-1)*t1*P,"P**(-1)*t2*P:=", P**(-1)*t2*P
              ,"P**(-1)*t3*P:=", P**(-1)*t3*P,"P**(-1)*t4*P:=", P**(-1)*t4*P$
write "matrice des derivations dans cette base diagonalisante Y(1),...,Y(7):"$
off nat$
write "P**(-1)*MATD*P:=",P**(-1)*MATD*P$

% Si besoin seconde diagonalisation  ou trigonalisation
matrix Q(7,7)$
Q:=ID$  %si pas besoin de deuxieme diagonalisation
%for i:=1:7 do <<Q(i,i):=1>>$   %Si pb pour diagonaliser t2 ou t3
%write "det Q:=",det(Q)$
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PP:=P*Q$
on nat$
write "PP:=",PP$
%write "PP**(-1)*t1*PP:=", PP**(-1)*t1*PP;
%write "PP**(-1)*t2*PP:=", PP**(-1)*t2*PP;
%write "PP**(-1)*t3*PP:=", PP**(-1)*t3*PP;

%write "matrice des derivations dans cette seconde base diagonalisante diaY(1),...,diaY(7):"$
%write "PP**(-1)*MATD*PP:=",PP**(-1)*MATD*PP$

MATDDIAGONALISE:=PP**(-1)*MATD*PP$
write "avec PP:=P*Q:=",PP$
write "MATDDIAGONALISE:=",MATDDIAGONALISE$

write "on voit apparaitre les poids sur la diagonale"$
for i:=1:7 do R(i):=MATDDIAGONALISE(i,i)$
for i:=1:7 do <<write R(i):=R(i)>>$

match matddiagonalise(6,6) =gamma3$
match matddiagonalise(3,3) =gamma1$
match matddiagonalise(1,1) =gamma2$
on factor$
for i:=1:7 do <<write R(i):=R(i)>>$
off factor$
weightsystem:={rkgtildedelta,9}$
%weightsystem:={0,5}$
%weightsystem:={1,{0,point,1} }$
write "Le systeme de poids est le systeme ",part(weightsystem,1),".",part(weightsystem,2)$

write "calcul de relations de commutation de la base diaY(j) diagonalisant le tore"$

listcommutateursdesX:=for i:=0:5 join for j:=i+1:6 collect {{i,j},X(i)*X(j)};

%changing the base X(0),... , X(6) to the diagonalizing one diaY(1), .. ,diaY(7)
operator diaY$
for j:=1:7 do diaY(j):=for k:=1:7 sum PP(k,j)*X(k-1)$
for j:=1:7 do write "diaY(",j,"):=",diaY(j)$

operator commutateursenXdesdiaY$
for i:=1:7 do for j:=1:7 do <<commutateursenXdesdiaY(i,j):=diaY(i)*diaY(j)>>$

listcommutateursenXdesdiaY:=for i:=1:6 join for j:=i+1:7 collect {{i,j},diaY(i)*diaY(j)}$
%write "listcommutateursenXdesdiaY:=",listcommutateursenXdesdiaY;


%changing the indexation of the X(i)s : passage a Y rebaptise YY    Y(j):=X(j-1)
%(decalage des indices)
operator YY$
noncomYY$
for j:=0:6 do X(j):=YY(j+1)$

listcommutateursenYYdesdiaY:=listcommutateursenXdesdiaY$
%write "listcommutateursenYYdesdiaY:=",listcommutateursenYYdesdiaY$
operator commutateursenYYdesdiaY$
for i:=1:7 do for j:=1:7 do <<commutateursenYYdesdiay(i,j):=commutateursenXdesdiaY(i,j)>>$


for j:=1:7 do <<clear diaY(j)>>$
for j:=0:6 do <<clear X(j)>>$

%writing the commutator list in terms of diaY(i)s instead of YY(i)s
%passage aux diaY
QQ:=PP**(-1)$
For j:=1:7 do YY(j):=for k:=1:7 sum QQ(k,j)*diaY(k)$

off nat$
write "liste des commutateurs des diaY(i) :"$
listcommutateursdesdiaY:=listcommutateursenYYdesdiaY$
operator commdiaY$
for i:=1:7 do for j:=1:7 do <<commdiay(i,j):=commutateursenYYdesdiaY(i,j)>>$
%write "liste totale des commutateursdesdiaY(i,j) :"$
%for i:=1:7 join for j:=1:7 collect {{i,j},commdiay(i,j)};
write "listcommutateurdesdiaY:=",listcommutateursdesdiaY;
for j:=1:7 do <<clear YY(j)>>$
%bye$
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%writing the file with the foregoing commutation  relations

PROCEDURE gtildedeltafile()$
begin$
off nat$
%a:=parama$  %%%%% if the parameter is a %%%%%
for j:=1:7 do diaY(j):=x(j)$
out "7nilpsgtildedelta/nilp6_53xC.IV"$  %fill in filename here
write "REFALGTEX:={nilp6_53xC,IV}"$     %fill in
write "dim:=7"$
write "operator x"$
write "noncom x"$
for each A in listcommutateursdesdiaY do
<<write if A neq 0 THEN "x(",part(part(A,1),1),")*x(",part(part(A,1),2),"):=",
part(A,2)>>$
write "for i:=1:7 do   <<x(i)*x(i):=0>>"$
write "for i:=1:6 do  for j:=i+1:7 do <<x(j)*x(i):=-x(i)*x(j)>>"$
write "operator v"$
write "noncom x,v"$
write "for i:=1:7 do  for j:=1:7 do <<v(j)*x(i):=if j=i then 1 else 0>>"$
write "for i:=1:7 do  R(i):=0"$
shut "7nilpsgtildedelta/nilp6_53xC.IV"$    %fill in
for j:=1:7 do <<clear diaY(j)>>$
end$


%gtildedeltafile()$   %dont forget to modify the files name above


%bye$
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write "Now we make explicit the isomorphism with an algebra of the book:"$
write "Namely g_{7,",part(weightsystem,1),".",part(weightsystem,2),"}"$
%write "(iL)"$
%write "(i)"$
%write "(iii)"$

%ws:=conditionssura$
%conditionssura:=append({0,-1},ws)$
%write "and that for a neq",conditionssura$

write "i.e. we go from the basis diaY(i) to the new basis ZZ(i) defined by the matrix:"$
write "on pose :"$
write "avec comme matrice de changement de base :"$

matrix isom(7,7)$
isom(3,1):=1$
isom(1,2):=1$
isom(6,3):=-1$
isom(7,4):=-1$
isom(2,5):=-1$
isom(5,6):=-1$
isom(4,7):=-1$

off nat$
write isom$
on factor$
write "det(isom):= ", det(isom)$
off factor$





%liste des commutateurs des ZZ(i) (ZZZ=ZZ:=
operator ZZ$
noncom ZZ$
on factor$
For j:=1:7 do ZZ(j):=for k:=1:7 sum isom(k,j)*diaY(k)$
For i:=1:7 do <<write "ZZ(",i,"):=",ZZ(i)>>$

%write "listcommutateurdesdiaY:=",listcommutateursdesdiaY;
listcommutateursdesZZendiaY:=
      for i:=1:6 join for j:=i+1:7 collect
      {{i,j},ws:=for k:=1:7 sum for l:=1:7 sum isom(k,i)*isom(l,j)*commdiay(k,l)}$
%write "listcommutateursdesZZendiaY:=",listcommutateursdesZZendiaY$


%formules d'inversion
inverseisom:=isom**(-1)$
for j:=1:7 do <<clear ZZ(j)>>$
For j:=1:7 do diaY(j):=for k:=1:7 sum inverseisom(k,j)*ZZ(k)$
listcommutateursdesZZ:=listcommutateursdesZZendiaY$
%write "listcommutateursdesZZ:=",listcommutateursdesZZ$
write "listcommutateursdesZZ:="$
off nat$
on factor$

for each W in listcommutateursdesZZ DO <<write {part(W,1),part(W,2)}>>$

write "We get the commutation relations of"$
write "g_{7,",part(weightsystem,1),".",part(weightsystem,2),"}"$

%write "(iL)"$
%write "(iii)"$
%write "(vi)"$
%off factor$
%write "with L:=",L$


%write "Et cela pour a:=",a,", b:=",b,"."$
write "Et cela pour a:=",a$
%write "and that for a neq ",conditionssura$

write "shortformdelta:=",shortformdelta$
write "delta:=",delta$
bye$
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write "The isomorphism from g_{7,",part(weightsystem,1),".",part(weightsystem,2),"} to gtildedelta"$
write "was constructed in 2 steps and is given by"$
write "the product matrix P*isom:=",P*isom$
write "which we record here under the name PSI"$
PSI:=P*isom$
out "psi6_.red"$  %fill in filename here
write "PSI6_:=",PSI$
shut "psi6_.red"$  %fill in filename here
bye$
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