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


        [0  0  0  0  0  0]
        [                ]
        [0  0  0  0  0  0]
        [                ]
        [0  0  0  0  0  0]
adx1 := [                ]
        [0  1  0  0  0  0]
        [                ]
        [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]
adx2 := [                 ]
        [-1  0  0  0  0  0]
        [                 ]
        [0   0  0  0  0  0]
        [                 ]
        [0   0  1  0  0  0]



        [0   0   0  0  0  0]
        [                  ]
        [0   0   0  0  0  0]
        [                  ]
        [0   0   0  0  0  0]
adx3 := [                  ]
        [0   0   0  0  0  0]
        [                  ]
        [-1  0   0  0  0  0]
        [                  ]
        [0   -1  0  0  0  0]


by subtracting adjoints one then may suppose:

xi(4,1):=0,xi(4,2):=0,xi(5,1):=0.

In the case 3, supposes :

xi(1,1):=0

xi(1,2):=0

xi(1,3):=0

xi(2,1):=0

xi(2,2):=0

xi(2,3):=0

xi(3,1):=0

xi(3,2):=0

xi(3,3):=0

delta:=

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


We denote this delta by the shortform

shortformdelta:={xi(4,3),

                 ss,

                 xi(5,2),

                 xi(5,3),

                 ss,

                 xi(6,1),

                 xi(6,2),

                 xi(6,3)}

paramindexeslist:={{4,3},{5,2},{5,3},{6,1},{6,2},{6,3}}

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

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

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

Unknown: d(3,6)
Unknown: d(3,6)
bonne inconnue W:=d(3,6)$

sa valeur doit etre WW:=0$

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

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

sa valeur doit etre WW:=d(3,1) - d(2,0)$

on resout l'equation {{0,1},5} qui est  maintenant AA:= - (d(5,6) + d(3,0))$

Unknowns: {d(5,6),d(3,0)}
Unknowns: {d(5,6),d(3,0)}
bonne inconnue W:=d(5,6)$

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

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

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

sa valeur doit etre WW:=d(1,1) + d(0,0)$

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

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

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

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

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

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

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

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

sa valeur doit etre WW:=0$

on resout l'equation {{0,3},4} 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 {{0,3},5} qui est  maintenant AA:= - d(5,4) + d(1,0)$

Unknowns: {d(5,4),d(1,0)}
Unknowns: {d(5,4),d(1,0)}
bonne inconnue W:=d(5,4)$

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

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

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

sa valeur doit etre WW:=d(2,0) + d(1,3)$

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

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

sa valeur doit etre WW:=0$

on resout l'equation {{0,5},6} 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 {{1,2},4} qui est  maintenant AA:= - d(3,3) + d(2,2) + d(1,
1) - d(0,0)$

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

sa valeur doit etre WW:=d(2,2) + d(1,1) - d(0,0)$

on resout l'equation {{1,2},5} qui est  maintenant AA:= - 2*d(1,0)$

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

sa valeur doit etre WW:=0$

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

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

sa valeur doit etre WW:= - (d(2,0) + d(1,3) + d(0,2))$

on resout l'equation {{1,3},0} 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$

on resout l'equation {{1,3},2} 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 {{1,3},4} qui est  maintenant AA:= - d(4,5) + d(2,3) + d(0,
1)$

Unknowns: {d(4,5),d(2,3),d(0,1)}
Unknowns: {d(4,5),d(2,3),d(0,1)}
bonne inconnue W:=d(4,5)$

sa valeur doit etre WW:=d(2,3) + d(0,1)$

on resout l'equation {{1,3},5} qui est  maintenant AA:= - d(5,5) + d(2,2) + 2*d(
1,1) - d(0,0)$

Unknowns: {d(5,5),d(2,2),d(1,1),d(0,0)}
Unknowns: {d(5,5),d(2,2),d(1,1),d(0,0)}
bonne inconnue W:=d(5,5)$

sa valeur doit etre WW:=d(2,2) + 2*d(1,1) - d(0,0)$

on resout l'equation {{1,3},6} qui est  maintenant AA:= - d(6,5) + d(2,1) - d(0,
3)$

Unknowns: {d(6,5),d(2,1),d(0,3)}
Unknowns: {d(6,5),d(2,1),d(0,3)}
bonne inconnue W:=d(6,5)$

sa valeur doit etre WW:=d(2,1) - d(0,3)$

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

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

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

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

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

sa valeur doit etre WW:=0$

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

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

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

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

Il n'y a pas de phase 2$

derivation generique de gtildedelta:$

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

pour delta:=

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


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

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

dim Der(gtildedelta):=20$

t1:=D(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  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]


Unknowns: {d(6,2),

           d(6,0),

           d(5,3),

           d(5,1),

           d(4,2),

           d(4,0),

           d(1,3),

           d(1,1),

           d(0,2),

           d(0,0)}

Unknowns: {d(6,2),

           d(6,0),

           d(5,3),

           d(5,1),

           d(4,2),

           d(4,0),

           d(1,3),

           d(1,1),

           d(0,2),

           d(0,0)}

commutant de t1 dans der(gtildedelta):


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

    (0,d(1,1),0,d(1,3),0,0,0),

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

    (0, - d(1,3),0,d(1,1),0,0,0),

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

    (0,d(5,1),0,d(5,3),0,2*d(1,1),0),

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


Unknowns: {d(6,2),

           d(6,0),

           d(5,3),

           d(5,1),

           d(4,2),

           d(4,0),

           d(1,3),

           d(1,1),

           d(0,2),

           d(0,0)}

Unknowns: {d(6,2),

           d(6,0),

           d(5,3),

           d(5,1),

           d(4,2),

           d(4,0),

           d(1,3),

           d(1,1),

           d(0,2),

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


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


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

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

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

t3:=D(0,2):=

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


t31:=calcul_t3(0,2)$

Unknowns: {d(1,3),d(1,1),d(0,2),d(0,0)}
Unknowns: {d(1,3),d(1,1),d(0,2),d(0,0)}
t3:=D(1,3):=

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


t32:=calcul_t3(1,3)$

t3:=-t31+t32$

t4:=4t31+t32$

t3:=
mat((0,0,-1,0,0,0,0),(0,0,0,1,0,0,0),(1,0,0,0,0,0,0),(0,-1,0,0,0,0,0),(0,0,0,0,0
,0,-2),(0,0,0,0,0,0,0),(0,0,0,0,2,0,0))$
$

t4:=
mat((0,0,4,0,0,0,0),(0,0,0,1,0,0,0),(-4,0,0,0,0,0,0),(0,-1,0,0,0,0,0),(0,0,0,0,0
,0,3),(0,0,0,0,0,0,0),(0,0,0,0,-3,0,0))$
$

t4*t3-t3*t4:=
mat((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),(0,0,0,0,0,0
,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0))$
$

rkgtildedelta:=4$

{{x,1,

  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [      0       ]
  [              ]
  [arbcomplex(55)]
  [              ]
  [      0       ]

  },

   2
 {x  + 1,

  2,


  [  - arbcomplex(56) ]
  [-------------------]
  [         x         ]
  [                   ]
  [  arbcomplex(57)   ]
  [ ----------------  ]
  [        x          ]
  [                   ]
  [  arbcomplex(56)   ]
  [                   ]
  [  arbcomplex(57)   ]
  [                   ]
  [         0         ]
  [                   ]
  [         0         ]
  [                   ]
  [         0         ]

  },

   2
 {x  + 4,

  1,


  [        0         ]
  [                  ]
  [        0         ]
  [                  ]
  [        0         ]
  [                  ]
  [        0         ]
  [                  ]
  [ arbcomplex(58)*x ]
  [------------------]
  [        2         ]
  [                  ]
  [        0         ]
  [                  ]
  [  arbcomplex(58)  ]

  }}

rank 4 with maximal torus t1,t2,t3,t4

4


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


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

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

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

[ - i  0  0   0      0     0   0 ]
[                                ]
[ 0    i  0   0      0     0   0 ]
[                                ]
[ 0    0  i   0      0     0   0 ]
[                                ]
[ 0    0  0   - i    0     0   0 ]
[                                ]
[ 0    0  0   0     - 2*i  0   0 ]
[                                ]
[ 0    0  0   0      0     0   0 ]
[                                ]
[ 0    0  0   0      0     0  2*i]

P**(-1)*t4*P:=

[4*i  0    0      0     0   0    0   ]
[                                    ]
[ 0   i    0      0     0   0    0   ]
[                                    ]
[ 0   0   - 4*i   0     0   0    0   ]
[                                    ]
[ 0   0    0      - i   0   0    0   ]
[                                    ]
[ 0   0    0      0    3*i  0    0   ]
[                                    ]
[ 0   0    0      0     0   0    0   ]
[                                    ]
[ 0   0    0      0     0   0   - 3*i]


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

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

PP:=

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


avec PP:=P*Q:=

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


MATDDIAGONALISE:=

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

      d(0,3) + i*d(0,1) + d(2,1) - i*d(2,3)
     ---------------------------------------,0,0,0),
                        2

    (0,i*d(1,3) + d(1,1),0,0,0,0,0),

        d(0,3) - i*d(0,1) + d(2,1) + i*d(2,3)
    (0,---------------------------------------, - (i*d(0,2) - d(0,0)),
                          2

      i*(d(2,1) - d(0,3) - i*d(0,1)) + d(2,3)
     -----------------------------------------,0,0,0),
                         2

    (0,0,0, - (i*d(1,3) - d(1,1)),0,0,0),

       - (i*(d(6,0) - d(4,2) + i*d(4,0)) - d(6,2))
    (----------------------------------------------,
                           2

       - (i*(d(6,1) - d(4,3) + i*d(4,1)) - d(6,3))
     ----------------------------------------------,
                           2

      d(4,2) + i*d(4,0) + d(6,0) - i*d(6,2)
     ---------------------------------------,
                        2

      d(4,3) + i*d(4,1) + d(6,1) - i*d(6,3)
     ---------------------------------------,
                        2

      - (i*(d(1,3) - d(0,2)) - (d(1,1) + d(0,0))),

      d(2,3) + d(0,1) - i*(d(2,1) - d(0,3))
     ---------------------------------------,0),
                        2

    (i*d(5,2) + d(5,0),i*d(5,3) + d(5,1),d(5,2) + i*d(5,0),d(5,3) + i*d(5,1),0,

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

      d(4,2) - i*d(4,0) + d(6,0) + i*d(6,2)
    (---------------------------------------,
                        2

      d(4,3) - i*d(4,1) + d(6,1) + i*d(6,3)
     ---------------------------------------,
                        2

      i*(d(6,0) - d(4,2) - i*d(4,0)) + d(6,2)
     -----------------------------------------,
                         2

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

       - (i*(d(2,3) + d(0,1)) - (d(2,1) - d(0,3)))
     ----------------------------------------------,
                           2

     i*(d(1,3) - d(0,2)) + d(1,1) + d(0,0)))


on voit apparaitre les poids sur la diagonale

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

r(2) := i*d(1,3) + d(1,1)

r(3) :=  - (i*d(0,2) - d(0,0))

r(4) :=  - (i*d(1,3) - d(1,1))

r(5) :=  - (i*(d(1,3) - d(0,2)) - (d(1,1) + d(0,0)))

r(6) := 2*d(1,1)

r(7) := i*(d(1,3) - d(0,2)) + d(1,1) + d(0,0)

r(1) := gamma2

r(2) := gamma3

r(3) := gamma4

r(4) := gamma1

r(5) := gamma1 + gamma2

r(6) := gamma1 + gamma3

r(7) := gamma3 + gamma4

Le systeme de poids est le systeme 4.1

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

listcommutateursdesx := {{{0,1},x(6)},

   {{0,2},0},

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

   {{0,4},0},

   {{0,5},0},

   {{0,6},0},

   {{1,2},x(4)},

   {{1,3},x(5)},

   {{1,4},0},

   {{1,5},0},

   {{1,6},0},

   {{2,3},x(6)},

   {{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):=i*x(2) + x(0)

diaY(2):=i*x(3) + x(1)

diaY(3):=x(2) + i*x(0)

diaY(4):=x(3) + i*x(1)

diaY(5):=i*x(6) + x(4)

diaY(6):=x(5)

diaY(7):=x(6) + i*x(4)

liste des commutateurs des diaY(i) :$

listcommutateurdesdiaY:={{{1,2},0},
{{1,3},0},
{{1,4},2*diay(5)},
{{1,5},0},
{{1,6},0},
{{1,7},0},
{{2,3}, - 2*i*diay(7)},
{{2,4},2*diay(6)},
{{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_{7,4.1}$

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((0,1,0,0,0,0,0),(0,0,1,0,0,0,0),(0,0,0,1,0,0,0),(1,0,0,0,0,0,0),(0,0,0,0,-2,
0,0),(0,0,0,0,0,-2,0),(0,0,0,0,0,0, - 2*i))$
$

det(isom):= 8*i$

ZZ(1):=diay(4)$

ZZ(2):=diay(1)$

ZZ(3):=diay(2)$

ZZ(4):=diay(3)$

ZZ(5):= - 2*diay(5)$

ZZ(6):= - 2*diay(6)$

ZZ(7):= - 2*i*diay(7)$

listcommutateursdesZZ:=$

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

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

{{1,4},0}$

{{1,5},0}$

{{1,6},0}$

{{1,7},0}$

{{2,3},0}$

{{2,4},0}$

{{2,5},0}$

{{2,6},0}$

{{2,7},0}$

{{3,4},zz(7)}$

{{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_{7,4.1}$

Et cela pour a:=1$

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

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

The isomorphism from g_{7,rkgtildedelta.1} to gtildedelta$

was constructed in 2 steps and is given by$

the product matrix P*isom:=
mat((0,1,0,i,0,0,0),(i,0,1,0,0,0,0),(0,i,0,1,0,0,0),(1,0,i,0,0,0,0),(0,0,0,0,-2,
0,2),(0,0,0,0,0,-2,0),(0,0,0,0, - 2*i,0, - 2*i))$
$

which we record here under the name PSI$

PSIcase3III2:=
mat((0,1,0,i,0,0,0),(i,0,1,0,0,0,0),(0,i,0,1,0,0,0),(1,0,i,0,0,0,0),(0,0,0,0,-2,
0,2),(0,0,0,0,0,-2,0),(0,0,0,0, - 2*i,0, - 2*i))$
$