rcalculderivgtildedelta6_nxC3case2I.r$

generic derivation : 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(2,2)
 + xi(1,1),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)))$

        [0  0  0  0  0  0]
        [                ]
        [0  0  0  0  0  0]
        [                ]
        [0  1  0  0  0  0]
adx1 := [                ]
        [0  0  0  0  0  0]
        [                ]
        [0  0  0  0  0  0]
        [                ]
        [0  0  0  0  0  0]


        [0   0  0  0  0  0]
        [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [-1  0  0  0  0  0]
adx2 := [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [0   0  0  0  0  0]


The generic nilpotent derivation : 

The matrices MATA:=((xi(1,1),xi(1,2)),(xi(2,1),xi(2,2))) is nilpotent


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

(WE denote MATA,MATB, since B is already for the entries of phi)

***We consider here the case 2 where MATA = 0 and MATB has nilpotent order 1.***

xi(1,1):= 0

xi(1,2):= 0

xi(2,1):= 0

xi(2,2):= 0

MATB:= 

[0  1  0]
[       ]
[0  0  0]
[       ]
[0  0  0]


MATC:= 

[xi(3,4)]
[       ]
[xi(3,5)]
[       ]
[xi(3,6)]


MATD:= 

[xi(4,1)  xi(4,2)]
[                ]
[xi(5,1)  xi(5,2)]
[                ]
[xi(6,1)  xi(6,2)]


by subtracting adjoints one then may suppose:

xi(3,1):=0,xi(3,2):=0.

delta:=

[   0        0     0     0        0        0   ]
[                                              ]
[   0        0     0     0        0        0   ]
[                                              ]
[   0        0     0  xi(3,4)  xi(3,5)  xi(3,6)]
[                                              ]
[xi(4,1)  xi(4,2)  0     0        1        0   ]
[                                              ]
[xi(5,1)  xi(5,2)  0     0        0        0   ]
[                                              ]
[xi(6,1)  xi(6,2)  0     0        0        0   ]


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}}

delta:=

[0  0  0  0  0  0]
[                ]
[0  0  0  0  0  0]
[                ]
[0  0  0  1  0  0]
[                ]
[0  0  0  0  1  0]
[                ]
[0  0  0  0  0  0]
[                ]
[0  1  0  0  0  0]


shortformdelta:={0,0,ss,0,0,ss,0,1,ss,1,0,0}$

on resout l'equation {{0,1},3} qui est  maintenant AA:=d(4,1) - d(2,0)$

Unknowns: {d(4,1),d(2,0)}
Unknowns: {d(4,1),d(2,0)}
bonne inconnue W:=d(4,1)$

sa valeur doit etre WW:=d(2,0)$

on resout l'equation {{0,1},4} qui est  maintenant AA:=d(5,1)$

Unknown: d(5,1)
Unknown: d(5,1)
bonne inconnue W:=d(5,1)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,1},6} qui est  maintenant AA:=d(2,1)$

Unknown: d(2,1)
Unknown: d(2,1)
bonne inconnue W:=d(2,1)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,2},0} qui est  maintenant AA:= - d(0,6)$

Unknown: d(0,6)
Unknown: d(0,6)
bonne inconnue W:=d(0,6)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,2},1} qui est  maintenant AA:= - d(1,6)$

Unknown: d(1,6)
Unknown: d(1,6)
bonne inconnue W:=d(1,6)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,2},2} qui est  maintenant AA:= - d(2,6)$

Unknown: d(2,6)
Unknown: d(2,6)
bonne inconnue W:=d(2,6)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,2},3} qui est  maintenant AA:=d(4,2) - d(3,6) + d(1,0)$

Unknowns: {d(4,2),d(3,6),d(1,0)}
Unknowns: {d(4,2),d(3,6),d(1,0)}
bonne inconnue W:=d(4,2)$

sa valeur doit etre WW:=d(3,6) - d(1,0)$

on resout l'equation {{0,2},4} qui est  maintenant AA:=d(5,2) - d(4,6)$

Unknowns: {d(5,2),d(4,6)}
Unknowns: {d(5,2),d(4,6)}
bonne inconnue W:=d(5,2)$

sa valeur doit etre WW:=d(4,6)$

on resout l'equation {{0,2},5} qui est  maintenant AA:= - d(5,6)$

Unknown: d(5,6)
Unknown: d(5,6)
bonne inconnue W:=d(5,6)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,2},6} qui est  maintenant AA:= - d(6,6) + d(2,2) + d(0,
0)$

Unknowns: {d(6,6),d(2,2),d(0,0)}
Unknowns: {d(6,6),d(2,2),d(0,0)}
bonne inconnue W:=d(6,6)$

sa valeur doit etre WW:=d(2,2) + d(0,0)$

on resout l'equation {{0,3},3} qui est  maintenant AA:=d(4,3)$

Unknown: d(4,3)
Unknown: d(4,3)
bonne inconnue W:=d(4,3)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,3},4} qui est  maintenant AA:=d(5,3)$

Unknown: d(5,3)
Unknown: d(5,3)
bonne inconnue W:=d(5,3)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,3},6} qui est  maintenant AA:=d(2,3)$

Unknown: d(2,3)
Unknown: d(2,3)
bonne inconnue W:=d(2,3)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,4},0} qui est  maintenant AA:= - d(0,3)$

Unknown: d(0,3)
Unknown: d(0,3)
bonne inconnue W:=d(0,3)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,4},1} qui est  maintenant AA:= - d(1,3)$

Unknown: d(1,3)
Unknown: d(1,3)
bonne inconnue W:=d(1,3)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,4},3} qui est  maintenant AA:=d(4,4) - d(3,3) + d(0,0)$

Unknowns: {d(4,4),d(3,3),d(0,0)}
Unknowns: {d(4,4),d(3,3),d(0,0)}
bonne inconnue W:=d(4,4)$

sa valeur doit etre WW:=d(3,3) - d(0,0)$

on resout l'equation {{0,4},4} qui est  maintenant AA:=d(5,4)$

Unknown: d(5,4)
Unknown: d(5,4)
bonne inconnue W:=d(5,4)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,4},6} qui est  maintenant AA:= - d(6,3) + d(2,4)$

Unknowns: {d(6,3),d(2,4)}
Unknowns: {d(6,3),d(2,4)}
bonne inconnue W:=d(6,3)$

sa valeur doit etre WW:=d(2,4)$

on resout l'equation {{0,5},0} qui est  maintenant AA:= - d(0,4)$

Unknown: d(0,4)
Unknown: d(0,4)
bonne inconnue W:=d(0,4)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,5},1} qui est  maintenant AA:= - d(1,4)$

Unknown: d(1,4)
Unknown: d(1,4)
bonne inconnue W:=d(1,4)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,5},2} qui est  maintenant AA:= - d(2,4)$

Unknown: d(2,4)
Unknown: d(2,4)
bonne inconnue W:=d(2,4)$

sa valeur doit etre WW:=0$

on resout l'equation {{0,5},3} qui est  maintenant AA:=d(4,5) - d(3,4)$

Unknowns: {d(4,5),d(3,4)}
Unknowns: {d(4,5),d(3,4)}
bonne inconnue W:=d(4,5)$

sa valeur doit etre WW:=d(3,4)$

on resout l'equation {{0,5},4} qui est  maintenant AA:=d(5,5) - d(3,3) + 2*d(0,0
)$

Unknowns: {d(5,5),d(3,3),d(0,0)}
Unknowns: {d(5,5),d(3,3),d(0,0)}
bonne inconnue W:=d(5,5)$

sa valeur doit etre WW:=d(3,3) - 2*d(0,0)$

on resout l'equation {{0,5},6} qui est  maintenant AA:= - d(6,4) + d(2,5)$

Unknowns: {d(6,4),d(2,5)}
Unknowns: {d(6,4),d(2,5)}
bonne inconnue W:=d(6,4)$

sa valeur doit etre WW:=d(2,5)$

on resout l'equation {{0,6},3} qui est  maintenant AA:=d(4,6)$

Unknown: d(4,6)
Unknown: d(4,6)
bonne inconnue W:=d(4,6)$

sa valeur doit etre WW:=0$

on resout l'equation {{1,2},3} qui est  maintenant AA:= - d(3,3) + d(2,2) + d(1,
1)$

Unknowns: {d(3,3),d(2,2),d(1,1)}
Unknowns: {d(3,3),d(2,2),d(1,1)}
bonne inconnue W:=d(3,3)$

sa valeur doit etre WW:=d(2,2) + d(1,1)$

on resout l'equation {{1,2},6} qui est  maintenant AA:=d(0,1)$

Unknown: d(0,1)
Unknown: d(0,1)
bonne inconnue W:=d(0,1)$

sa valeur doit etre WW:=0$

on resout l'equation {{1,5},3} qui est  maintenant AA:=d(2,5)$

Unknown: d(2,5)
Unknown: d(2,5)
bonne inconnue W:=d(2,5)$

sa valeur doit etre WW:=0$

on resout l'equation {{2,4},3} qui est  maintenant AA:=d(0,2)$

Unknown: d(0,2)
Unknown: d(0,2)
bonne inconnue W:=d(0,2)$

sa valeur doit etre WW:=0$

on resout l'equation {{2,5},3} qui est  maintenant AA:= - d(1,5)$

Unknown: d(1,5)
Unknown: d(1,5)
bonne inconnue W:=d(1,5)$

sa valeur doit etre WW:=0$

on resout l'equation {{2,5},6} qui est  maintenant AA:= - d(0,5)$

Unknown: d(0,5)
Unknown: d(0,5)
bonne inconnue W:=d(0,5)$

sa valeur doit etre WW:=0$

Derivation equations to cancel (Reduce output) : \\{{{{0,1},3},0},
{{{0,1},4},0},
{{{0,1},6},0},
{{{0,2},0},0},
{{{0,2},1},0},
{{{0,2},2},0},
{{{0,2},3},0},
{{{0,2},4},0},
{{{0,2},5},0},
{{{0,2},6},0},
{{{0,3},3},0},
{{{0,3},4},0},
{{{0,3},6},0},
{{{0,4},0},0},
{{{0,4},1},0},
{{{0,4},2},0},
{{{0,4},3},0},
{{{0,4},4},0},
{{{0,4},5},0},
{{{0,4},6},0},
{{{0,5},0},0},
{{{0,5},1},0},
{{{0,5},2},0},
{{{0,5},3},0},
{{{0,5},4},0},
{{{0,5},5},0},
{{{0,5},6},0},
{{{0,6},3},0},
{{{0,6},4},0},
{{{0,6},6},0},
{{{1,2},0},0},
{{{1,2},1},0},
{{{1,2},2},0},
{{{1,2},3},0},
{{{1,2},4},0},
{{{1,2},5},0},
{{{1,2},6},0},
{{{1,3},3},0},
{{{1,4},3},0},
{{{1,5},3},0},
{{{1,5},4},0},
{{{1,6},3},0},
{{{2,3},3},0},
{{{2,3},6},0},
{{{2,4},3},0},
{{{2,4},6},0},
{{{2,5},3},0},
{{{2,5},4},0},
{{{2,5},6},0},
{{{2,6},3},0},
{{{2,6},6},0},
{{{3,4},3},0},
{{{3,5},4},0},
{{{4,5},3},0},
{{{4,5},4},0},
{{{4,6},3},0},
{{{5,6},4},0}}$

Il n'y a pas de phase 2$

derivation generique de gtildedelta:$

MATD:=
mat((d(0,0),0,0,0,0,0,0),(d(1,0),d(1,1),d(1,2),0,0,0,0),(d(2,0),0,d(2,2),0,0,0,0
),(d(3,0),d(3,1),d(3,2),d(2,2) + d(1,1),d(3,4),d(3,5),d(3,6)),(d(4,0),d(2,0),d(3
,6) - d(1,0),0,d(2,2) + d(1,1) - d(0,0),d(3,4),0),(d(5,0),0,0,0,0,d(2,2) + d(1,1
) - 2*d(0,0),0),(d(6,0),d(6,1),d(6,2),0,0,d(6,5),d(2,2) + d(0,0)))$

pour delta:=

[0  0  0  0  0  0]
[                ]
[0  0  0  0  0  0]
[                ]
[0  0  0  1  0  0]
[                ]
[0  0  0  0  1  0]
[                ]
[0  0  0  0  0  0]
[                ]
[0  1  0  0  0  0]


pour shortformdelta:={0,0,ss,0,0,ss,0,1,ss,1,0,0}

Unknowns: {d(6,5),
d(6,2),
d(6,1),
d(6,0),
d(5,0),
d(4,0),
d(3,6),
d(3,5),
d(3,4),
d(3,2),
d(3,1),
d(3,0),
d(2,2),
d(2,0),
d(1,2),
d(1,1),
d(1,0),
d(0,0)}
Unknowns: {d(6,5),
d(6,2),
d(6,1),
d(6,0),
d(5,0),
d(4,0),
d(3,6),
d(3,5),
d(3,4),
d(3,2),
d(3,1),
d(3,0),
d(2,2),
d(2,0),
d(1,2),
d(1,1),
d(1,0),
d(0,0)}
listeparametresMATD{d(6,5),
d(6,2),
d(6,1),
d(6,0),
d(5,0),
d(4,0),
d(3,6),
d(3,5),
d(3,4),
d(3,2),
d(3,1),
d(3,0),
d(2,2),
d(2,0),
d(1,2),
d(1,1),
d(1,0),
d(0,0)}$

dim Der(gtildedelta):=18$

t1:=D(0,0):=

[1  0  0  0  0   0   0]
[                     ]
[0  0  0  0  0   0   0]
[                     ]
[0  0  0  0  0   0   0]
[                     ]
[0  0  0  0  0   0   0]
[                     ]
[0  0  0  0  -1  0   0]
[                     ]
[0  0  0  0  0   -2  0]
[                     ]
[0  0  0  0  0   0   1]


Unknowns: {d(6,0),d(3,2),d(3,1),d(2,2),d(1,2),d(1,1),d(0,0)}

Unknowns: {d(6,0),d(3,2),d(3,1),d(2,2),d(1,2),d(1,1),d(0,0)}

t2:=D(1,1):=

[0  0  0  0  0  0  0]
[                   ]
[0  1  0  0  0  0  0]
[                   ]
[0  0  0  0  0  0  0]
[                   ]
[0  0  0  1  0  0  0]
[                   ]
[0  0  0  0  1  0  0]
[                   ]
[0  0  0  0  0  1  0]
[                   ]
[0  0  0  0  0  0  0]


Unknowns: {d(6,0),d(3,1),d(2,2),d(1,1),d(0,0)}
Unknowns: {d(6,0),d(3,1),d(2,2),d(1,1),d(0,0)}
commutant simultane de t1,t2 dans der(gtildedelta):$

mat((d(0,0),0,0,0,0,0,0),(0,d(1,1),0,0,0,0,0),(0,0,d(2,2),0,0,0,0),(0,d(3,1),0,d
(2,2) + d(1,1),0,0,0),(0,0,0,0,d(1,1) - d(0,0) + d(2,2),0,0),(0,0,0,0,0,d(1,1) -
 2*d(0,0) + d(2,2),0),(d(6,0),0,0,0,0,0,d(2,2) + d(0,0)))$

Unknowns: {d(6,0),d(3,1),d(2,2),d(1,1),d(0,0)}

Unknowns: {d(6,0),d(3,1),d(2,2),d(1,1),d(0,0)}

t3:=D(2,2):=

[0  0  0  0  0  0  0]
[                   ]
[0  0  0  0  0  0  0]
[                   ]
[0  0  1  0  0  0  0]
[                   ]
[0  0  0  1  0  0  0]
[                   ]
[0  0  0  0  1  0  0]
[                   ]
[0  0  0  0  0  1  0]
[                   ]
[0  0  0  0  0  0  1]


Unknowns: {d(2,2),d(1,1),d(0,0)}

Unknowns: {d(2,2),d(1,1),d(0,0)}

commutant simultane de t1,t2,t3 dans der(gtildedelta):

mat((d(0,0),0,0,0,0,0,0),

    (0,d(1,1),0,0,0,0,0),

    (0,0,d(2,2),0,0,0,0),

    (0,0,0,d(2,2) + d(1,1),0,0,0),

    (0,0,0,0,d(1,1) - d(0,0) + d(2,2),0,0),

    (0,0,0,0,0,d(1,1) - 2*d(0,0) + d(2,2),0),

    (0,0,0,0,0,0,d(2,2) + d(0,0)))


rank 3 with maximal torus t1,t2,t3

3


matrice de passage de la base X(0)=delta, X(1),..., X(6) a une base  diagonali\
sant le tore maximal: on peut prendre

P:=

[1  0  0  0  0  0  0]
[                   ]
[0  1  0  0  0  0  0]
[                   ]
[0  0  1  0  0  0  0]
[                   ]
[0  0  0  1  0  0  0]
[                   ]
[0  0  0  0  1  0  0]
[                   ]
[0  0  0  0  0  1  0]
[                   ]
[0  0  0  0  0  0  1]


P**(-1)*t1*P:=

[1  0  0  0  0   0   0]
[                     ]
[0  0  0  0  0   0   0]
[                     ]
[0  0  0  0  0   0   0]
[                     ]
[0  0  0  0  0   0   0]
[                     ]
[0  0  0  0  -1  0   0]
[                     ]
[0  0  0  0  0   -2  0]
[                     ]
[0  0  0  0  0   0   1]

P**(-1)*t2*P:=

[0  0  0  0  0  0  0]
[                   ]
[0  1  0  0  0  0  0]
[                   ]
[0  0  0  0  0  0  0]
[                   ]
[0  0  0  1  0  0  0]
[                   ]
[0  0  0  0  1  0  0]
[                   ]
[0  0  0  0  0  1  0]
[                   ]
[0  0  0  0  0  0  0]

P**(-1)*t3*P:=

[0  0  0  0  0  0  0]
[                   ]
[0  0  0  0  0  0  0]
[                   ]
[0  0  1  0  0  0  0]
[                   ]
[0  0  0  1  0  0  0]
[                   ]
[0  0  0  0  1  0  0]
[                   ]
[0  0  0  0  0  1  0]
[                   ]
[0  0  0  0  0  0  1]


matrice des derivations dans cette base diagonalisante Y(1),...,Y(7):

P**(-1)*MATD*P:=
mat((d(0,0),0,0,0,0,0,0),(d(1,0),d(1,1),d(1,2),0,0,0,0),(d(2,0),0,d(2,2),0,0,0,0
),(d(3,0),d(3,1),d(3,2),d(2,2) + d(1,1),d(3,4),d(3,5),d(3,6)),(d(4,0),d(2,0),d(3
,6) - d(1,0),0,d(1,1) - d(0,0) + d(2,2),d(3,4),0),(d(5,0),0,0,0,0,d(1,1) - 2*d(0
,0) + d(2,2),0),(d(6,0),d(6,1),d(6,2),0,0,d(6,5),d(2,2) + d(0,0)))$

PP:=

[1  0  0  0  0  0  0]
[                   ]
[0  1  0  0  0  0  0]
[                   ]
[0  0  1  0  0  0  0]
[                   ]
[0  0  0  1  0  0  0]
[                   ]
[0  0  0  0  1  0  0]
[                   ]
[0  0  0  0  0  1  0]
[                   ]
[0  0  0  0  0  0  1]


avec PP:=P*Q:=

[1  0  0  0  0  0  0]
[                   ]
[0  1  0  0  0  0  0]
[                   ]
[0  0  1  0  0  0  0]
[                   ]
[0  0  0  1  0  0  0]
[                   ]
[0  0  0  0  1  0  0]
[                   ]
[0  0  0  0  0  1  0]
[                   ]
[0  0  0  0  0  0  1]


MATDDIAGONALISE:=

mat((d(0,0),0,0,0,0,0,0),

    (d(1,0),d(1,1),d(1,2),0,0,0,0),

    (d(2,0),0,d(2,2),0,0,0,0),

    (d(3,0),d(3,1),d(3,2),d(2,2) + d(1,1),d(3,4),d(3,5),d(3,6)),

    (d(4,0),d(2,0),d(3,6) - d(1,0),0,d(1,1) - d(0,0) + d(2,2),d(3,4),0),

    (d(5,0),0,0,0,0,d(1,1) - 2*d(0,0) + d(2,2),0),

    (d(6,0),d(6,1),d(6,2),0,0,d(6,5),d(2,2) + d(0,0)))


on voit apparaitre les poids sur la diagonale

*** r declared operator 

r(1) := d(0,0)

r(2) := d(1,1)

r(3) := d(2,2)

r(4) := d(2,2) + d(1,1)

r(5) := d(1,1) - d(0,0) + d(2,2)

r(6) := d(1,1) - 2*d(0,0) + d(2,2)

r(7) := d(2,2) + d(0,0)

r(1) := gamma1

r(2) :=  - (gamma2 - gamma3 - 2*gamma1)

r(3) := gamma2

r(4) := 2*gamma1 + gamma3

r(5) := gamma1 + gamma3

r(6) := gamma3

r(7) := gamma1 + gamma2

Le systeme de poids est le systeme 3.14

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

listcommutateursdesx := {{{0,1},0},

   {{0,2},x(6)},

   {{0,3},0},

   {{0,4},x(3)},

   {{0,5},x(4)},

   {{0,6},0},

   {{1,2},x(3)},

   {{1,3},0},

   {{1,4},0},

   {{1,5},0},

   {{1,6},0},

   {{2,3},0},

   {{2,4},0},

   {{2,5},0},

   {{2,6},0},

   {{3,4},0},

   {{3,5},0},

   {{3,6},0},

   {{4,5},0},

   {{4,6},0},

   {{5,6},0}}

diaY(1):=x(0)

diaY(2):=x(1)

diaY(3):=x(2)

diaY(4):=x(3)

diaY(5):=x(4)

diaY(6):=x(5)

diaY(7):=x(6)

liste des commutateurs des diaY(i) :$

listcommutateurdesdiaY:={{{1,2},0},
{{1,3},diay(7)},
{{1,4},0},
{{1,5},diay(4)},
{{1,6},diay(5)},
{{1,7},0},
{{2,3},diay(4)},
{{2,4},0},
{{2,5},0},
{{2,6},0},
{{2,7},0},
{{3,4},0},
{{3,5},0},
{{3,6},0},
{{3,7},0},
{{4,5},0},
{{4,6},0},
{{4,7},0},
{{5,6},0},
{{5,7},0},
{{6,7},0}}$

Now we make explicit the isomorphism with an algebra of the book:$

Namely g_{7,3.14}$

i.e. we go from the basis diaY(i) to the new basis ZZ(i) defined by the matrix:$

on pose :$

avec comme matrice de changement de base :$

mat((1,0,0,0,0,0,0),(0,0,0,1,0,0,0),(0,1,0,0,0,0,0),(0,0,0,0,0,0,-1),(0,0,0,0,0,
-1,0),(0,0,-1,0,0,0,0),(0,0,0,0,1,0,0))$

det(isom):= 1$

ZZ(1):=diay(1)$

ZZ(2):=diay(3)$

ZZ(3):= - diay(6)$

ZZ(4):=diay(2)$

ZZ(5):=diay(7)$

ZZ(6):= - diay(5)$

ZZ(7):= - diay(4)$

listcommutateursdesZZ:=$

{{1,2},zz(5)}$

{{1,3},zz(6)}$

{{1,4},0}$

{{1,5},0}$

{{1,6},zz(7)}$

{{1,7},0}$

{{2,3},0}$

{{2,4},zz(7)}$

{{2,5},0}$

{{2,6},0}$

{{2,7},0}$

{{3,4},0}$

{{3,5},0}$

{{3,6},0}$

{{3,7},0}$

{{4,5},0}$

{{4,6},0}$

{{4,7},0}$

{{5,6},0}$

{{5,7},0}$

{{6,7},0}$

We get the commutation relations of$

g_{7,3.14}$

shortformdelta:={0,0,ss,0,0,ss,0,1,ss,1,0,0}$

delta:=
mat((0,0,0,0,0,0),(0,0,0,0,0,0),(0,0,0,1,0,0),(0,0,0,0,1,0),(0,0,0,0,0,0),(0,1,0
,0,0,0))$