generic derivation : delta:= mat((xi(1,1),0,0,0,0,0),(xi(2,1),xi(2,2),0,0,0,0),(xi(3,1),xi(3,2),2*xi(1,1),0,0 ,0),(xi(4,1),xi(4,2), - xi(2,1),xi(2,2) + xi(1,1),0,0),(xi(5,1),xi(5,2),xi(5,3), xi(4,2) - xi(3,1),xi(2,2) + 2*xi(1,1),xi(5,6)),(xi(6,1),xi(6,2),xi(6,3),0,0,xi(6 ,6)))$ $ [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 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] adx2 := [ ] [-1 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] [ ] [0 0 0 0 0 0] adx3 := [ ] [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 0] [ ] [0 0 0 0 0 0] adx4 := [ ] [0 0 0 0 0 0] [ ] [-1 0 0 0 0 0] [ ] [0 0 0 0 0 0] The generic nilpotent derivation : the eigenvalues are 0 xi(1,1):=0 xi(2,2):=0 xi(6,6):=0 by subtracting adjoints one then may suppose: xi(4,1):=0,xi(4,2):=0,xi(5,1):=0,xi(5,2):=0 delta:= [ 0 0 0 0 0 0 ] [ ] [xi(2,1) 0 0 0 0 0 ] [ ] [xi(3,1) xi(3,2) 0 0 0 0 ] [ ] [ 0 0 - xi(2,1) 0 0 0 ] [ ] [ 0 0 xi(5,3) - xi(3,1) 0 xi(5,6)] [ ] [xi(6,1) xi(6,2) xi(6,3) 0 0 0 ] We denote this delta by the shortform shortformdelta:={xi(2,1), ss, xi(3,1), xi(3,2), ss, xi(5,3), xi(5,6), ss, xi(6,1), xi(6,2), xi(6,3)} paramindexeslist:={{2,1},{3,1},{3,2},{5,3},{5,6},{6,1},{6,2},{6,3}} a neq {}$ a:=a$ delta:= mat((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,a,0,0,1),(1,0,0 ,0,0,0))$ $ shortformdelta:={0, ss, 0, 1, ss, a, 1, 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) + d(2,1)$ Unknowns: {d(3,6),d(2,1)} Unknowns: {d(3,6),d(2,1)} bonne inconnue W:=d(3,6)$ sa valeur doit etre WW:=d(2,1)$ on resout l'equation {{0,1},4} qui est maintenant AA:= - (d(4,6) + d(2,0))$ Unknowns: {d(4,6),d(2,0)} Unknowns: {d(4,6),d(2,0)} bonne inconnue W:=d(4,6)$ sa valeur doit etre WW:= - d(2,0)$ on resout l'equation {{0,1},5} qui est maintenant AA:=d(6,1) - d(5,6) - d(4,0) + d(3,1)*a$ Unknowns: {d(6,1),d(5,6),d(4,0),d(3,1),a} Unknowns: {d(6,1),d(5,6),d(4,0),d(3,1),a} bonne inconnue W:=d(6,1)$ sa valeur doit etre WW:=d(5,6) + d(4,0) - d(3,1)*a$ 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},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 {{0,2},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 {{0,2},2} 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 {{0,2},3} qui est maintenant AA:= - d(3,3) + d(2,2) + d(0, 0)$ Unknowns: {d(3,3),d(2,2),d(0,0)} Unknowns: {d(3,3),d(2,2),d(0,0)} bonne inconnue W:=d(3,3)$ sa valeur doit etre WW:=d(2,2) + d(0,0)$ on resout l'equation {{0,2},4} qui est maintenant AA:= - d(4,3) + d(1,0)$ Unknowns: {d(4,3),d(1,0)} Unknowns: {d(4,3),d(1,0)} bonne inconnue W:=d(4,3)$ sa valeur doit etre WW:=d(1,0)$ on resout l'equation {{0,2},5} qui est maintenant AA:=d(6,2) - d(5,3) + d(3,2)* a - d(3,0)$ Unknowns: {d(6,2),d(5,3),d(3,2),d(3,0),a} Unknowns: {d(6,2),d(5,3),d(3,2),d(3,0),a} bonne inconnue W:=d(6,2)$ sa valeur doit etre WW:=d(5,3) - d(3,2)*a + d(3,0)$ on resout l'equation {{0,2},6} qui est maintenant AA:= - d(6,3) + d(1,2)$ Unknowns: {d(6,3),d(1,2)} Unknowns: {d(6,3),d(1,2)} bonne inconnue W:=d(6,3)$ sa valeur doit etre WW:=d(1,2)$ on resout l'equation {{0,3},0} qui est maintenant AA:= - d(0,5)*a$ Unknowns: {d(0,5),a} Unknowns: {d(0,5),a} pas de selection possible de variable a coefficient numerique dans - d(0,5)*a on resout l'equation {{0,3},1} qui est maintenant AA:= - d(1,5)*a$ Unknowns: {d(1,5),a} Unknowns: {d(1,5),a} pas de selection possible de variable a coefficient numerique dans - d(1,5)*a on resout l'equation {{0,3},2} qui est maintenant AA:= - d(2,5)*a$ Unknowns: {d(2,5),a} Unknowns: {d(2,5),a} pas de selection possible de variable a coefficient numerique dans - d(2,5)*a on resout l'equation {{0,3},3} qui est maintenant AA:= - d(3,5)*a$ Unknowns: {d(3,5),a} Unknowns: {d(3,5),a} pas de selection possible de variable a coefficient numerique dans - d(3,5)*a on resout l'equation {{0,3},4} qui est maintenant AA:= - d(4,5)*a$ Unknowns: {d(4,5),a} Unknowns: {d(4,5),a} pas de selection possible de variable a coefficient numerique dans - d(4,5)*a on resout l'equation {{0,3},5} qui est maintenant AA:= - d(5,5)*a + d(2,2)*a + d(2,0) + d(1,2) + 2*d(0,0)*a$ Unknowns: {d(5,5),d(2,2),d(2,0),d(1,2),d(0,0),a} Unknowns: {d(5,5),d(2,2),d(2,0),d(1,2),d(0,0),a} bonne inconnue W:=d(1,2)$ sa valeur doit etre WW:=d(5,5)*a - d(2,2)*a - d(2,0) - 2*d(0,0)*a$ on resout l'equation {{0,3},6} qui est maintenant AA:= - d(6,5)*a$ Unknowns: {d(6,5),a} Unknowns: {d(6,5),a} pas de selection possible de variable a coefficient numerique dans - d(6,5)*a on resout l'equation {{0,4},3} 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 {{0,4},5} qui est maintenant AA:=d(6,4) + d(3,4)*a + d(1,0 )$ Unknowns: {d(6,4),d(3,4),d(1,0),a} Unknowns: {d(6,4),d(3,4),d(1,0),a} bonne inconnue W:=d(6,4)$ sa valeur doit etre WW:= - (d(3,4)*a + d(1,0))$ on resout l'equation {{0,4},6} 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},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 {{0,5},5} qui est maintenant AA:=d(6,5) + d(3,5)*a$ Unknowns: {d(6,5),d(3,5),a} Unknowns: {d(6,5),d(3,5),a} bonne inconnue W:=d(6,5)$ sa valeur doit etre WW:= - d(3,5)*a$ 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 {{0,6},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 {{0,6},3} 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 {{0,6},4} qui est maintenant AA:= - d(4,5)$ Unknown: d(4,5) Unknown: d(4,5) bonne inconnue W:=d(4,5)$ sa valeur doit etre WW:=0$ on resout l'equation {{0,6},5} qui est maintenant AA:= - d(5,5) + d(2,1)*a + d( 1,1) + 2*d(0,0)$ Unknowns: {d(5,5),d(2,1),d(1,1),d(0,0),a} Unknowns: {d(5,5),d(2,1),d(1,1),d(0,0),a} bonne inconnue W:=d(5,5)$ sa valeur doit etre WW:=d(2,1)*a + d(1,1) + 2*d(0,0)$ on resout l'equation {{1,2},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 {{1,2},3} qui est maintenant AA:= - d(3,4) + d(0,1)$ Unknowns: {d(3,4),d(0,1)} Unknowns: {d(3,4),d(0,1)} bonne inconnue W:=d(3,4)$ sa valeur doit etre WW:=d(0,1)$ on resout l'equation {{1,2},4} qui est maintenant AA:= - d(4,4) + d(2,2) + d(1, 1)$ Unknowns: {d(4,4),d(2,2),d(1,1)} Unknowns: {d(4,4),d(2,2),d(1,1)} bonne inconnue W:=d(4,4)$ sa valeur doit etre WW:=d(2,2) + d(1,1)$ on resout l'equation {{1,2},5} qui est maintenant AA:= - d(5,4) + d(4,2) - d(3, 1)$ Unknowns: {d(5,4),d(4,2),d(3,1)} Unknowns: {d(5,4),d(4,2),d(3,1)} bonne inconnue W:=d(5,4)$ sa valeur doit etre WW:=d(4,2) - d(3,1)$ on resout l'equation {{1,2},6} qui est maintenant AA:=d(1,0) - d(0,2) + d(0,1)* a$ Unknowns: {d(1,0),d(0,2),d(0,1),a} Unknowns: {d(1,0),d(0,2),d(0,1),a} bonne inconnue W:=d(1,0)$ sa valeur doit etre WW:=d(0,2) - d(0,1)*a$ on resout l'equation {{1,3},5} qui est maintenant AA:=d(2,1) + d(0,2)$ Unknowns: {d(2,1),d(0,2)} Unknowns: {d(2,1),d(0,2)} bonne inconnue W:=d(2,1)$ sa valeur doit etre WW:= - d(0,2)$ on resout l'equation {{1,4},5} qui est maintenant AA:=d(2,2) + d(1,1) + d(0,2)* a - 2*d(0,0)$ Unknowns: {d(2,2),d(1,1),d(0,2),d(0,0),a} Unknowns: {d(2,2),d(1,1),d(0,2),d(0,0),a} bonne inconnue W:=d(2,2)$ sa valeur doit etre WW:= - d(1,1) - d(0,2)*a + 2*d(0,0)$ on resout l'equation {{1,6},5} qui est maintenant AA:= - d(2,0) + d(0,1)$ Unknowns: {d(2,0),d(0,1)} Unknowns: {d(2,0),d(0,1)} bonne inconnue W:=d(0,1)$ sa valeur doit etre WW:=d(2,0)$ on resout l'equation {{2,3},5} qui est maintenant AA:=3*( - d(1,1) + d(0,0))$ Unknowns: {d(1,1),d(0,0)} Unknowns: {d(1,1),d(0,0)} bonne inconnue W:=d(1,1)$ 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},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},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},3},0}, {{{0,4},5},0}, {{{0,4},6},0}, {{{0,5},3},0}, {{{0,5},5},0}, {{{0,5},6},0}, {{{0,6},0},0}, {{{0,6},1},0}, {{{0,6},2},0}, {{{0,6},3},0}, {{{0,6},4},0}, {{{0,6},5},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},4},0}, {{{1,3},5},0}, {{{1,3},6},0}, {{{1,4},0},0}, {{{1,4},1},0}, {{{1,4},2},0}, {{{1,4},3},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},3},0}, {{{2,4},4},0}, {{{2,4},5},0}, {{{2,5},3},0}, {{{2,5},4},0}, {{{2,5},5},0}, {{{2,6},3},0}, {{{2,6},4},0}, {{{2,6},5},0}, {{{3,4},5},0}, {{{3,5},5},0}, {{{3,6},5},0}, {{{4,5},5},0}, {{{4,6},5},0}, {{{5,6},5},0}}$ Il n'y a pas de phase 2$ derivation generique de gtildedelta:$ MATD:= mat((d(0,0),d(2,0),d(0,2),0,0,0,0),( - d(2,0)*a + d(0,2),d(0,0), - d(2,0),0,0,0, 0),(d(2,0), - d(0,2), - d(0,2)*a + d(0,0),0,0,0,0),(d(3,0),d(3,1),d(3,2), - d(0, 2)*a + 2*d(0,0),d(2,0),0, - d(0,2)),(d(4,0),d(4,1),d(4,2), - d(2,0)*a + d(0,2), - d(0,2)*a + 2*d(0,0),0, - d(2,0)),(d(5,0),d(5,1),d(5,2),d(5,3),d(4,2) - d(3,1), - d(0,2)*a + 3*d(0,0),d(5,6)),(d(6,0),d(5,6) + d(4,0) - d(3,1)*a,d(5,3) - d(3,2 )*a + d(3,0), - d(2,0), - d(0,2),0,2*d(0,0)))$ $ pour delta:= [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 a 0 0 1] [ ] [1 0 0 0 0 0] pour shortformdelta:={0, ss, 0, 1, ss, a, 1, ss, 1, 0, 0} Unknowns: {d(6,0), d(5,6), d(5,3), d(5,2), d(5,1), d(5,0), d(4,2), d(4,1), d(4,0), d(3,2), d(3,1), d(3,0), d(2,0), d(0,2), d(0,0), a} Unknowns: {d(6,0), d(5,6), d(5,3), d(5,2), d(5,1), d(5,0), d(4,2), d(4,1), d(4,0), d(3,2), d(3,1), d(3,0), d(2,0), d(0,2), d(0,0), a} listeparametresMATD{d(6,0), d(5,6), d(5,3), d(5,2), d(5,1), d(5,0), d(4,2), d(4,1), d(4,0), d(3,2), d(3,1), d(3,0), d(2,0), d(0,2), d(0,0)}$ dim Der(gtildedelta):=15$ t1:=D(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 2 0 0 0] [ ] [0 0 0 0 2 0 0] [ ] [0 0 0 0 0 3 0] [ ] [0 0 0 0 0 0 2] Unknowns: {d(2,0),d(0,2),d(0,0),a} Unknowns: {d(2,0),d(0,2),d(0,0),a} commutant de t1 dans der(gtildedelta): mat((d(0,0),d(2,0),d(0,2),0,0,0,0), ( - d(2,0)*a + d(0,2),d(0,0), - d(2,0),0,0,0,0), (d(2,0), - d(0,2), - d(0,2)*a + d(0,0),0,0,0,0), (0,0,0, - d(0,2)*a + 2*d(0,0),d(2,0),0, - d(0,2)), (0,0,0, - d(2,0)*a + d(0,2), - d(0,2)*a + 2*d(0,0),0, - d(2,0)), (0,0,0,0,0, - d(0,2)*a + 3*d(0,0),0), (0,0,0, - d(2,0), - d(0,2),0,2*d(0,0))) Unknowns: {d(2,0),d(0,2),d(0,0),a} Unknowns: {d(2,0),d(0,2),d(0,0),a} t2:=D(0,2):= [0 0 1 0 0 0 0 ] [ ] [1 0 0 0 0 0 0 ] [ ] [0 -1 - a 0 0 0 0 ] [ ] [0 0 0 - a 0 0 -1] [ ] [0 0 0 1 - a 0 0 ] [ ] [0 0 0 0 0 - a 0 ] [ ] [0 0 0 0 -1 0 0 ] - (a**2*x + 2*a*x**2 + x**3 - 1)*(a*x**2 + x**3 + 1)*(a + x)$ {{a + x, 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(129)] [ ] [ 0 ] }, 2 3 {a*x + x + 1, 1, [ (a + 2*x)*(a + x)*arbcomplex(130) ] [ ----------------------------------- ] [ 2 2 ] [ a *x + a*x - 2 ] [ ] [ 3 2 2 ] [ - (a *x + a *x - 3*a - 2*x)*arbcomplex(130) ] [-----------------------------------------------] [ 2 2 ] [ a *x + a*x - 2 ] [ ] [ arbcomplex(130) ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }, 2 {(a + 2*x)*a*x + (x + x + 1)*(x - 1), 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ - (a + 2*x)*(a + x)*arbcomplex(131) ] [--------------------------------------] [ 3 2 2 ] [ a + 2*a *x + a*x + 2 ] [ ] [ - (a + 2*x)*arbcomplex(131) ] [ ------------------------------ ] [ 3 2 2 ] [ a + 2*a *x + a*x + 2 ] [ ] [ 0 ] [ ] [ arbcomplex(131) ] }} Unknowns: {d(2,0),d(0,2),d(0,0),a} Unknowns: {d(2,0),d(0,2),d(0,0),a} commutant simultane de t1,t2 dans der(gtildedelta):$ mat((d(0,0),d(2,0),d(0,2),0,0,0,0),( - (d(2,0)*a - d(0,2)),d(0,0), - d(2,0),0,0, 0,0),(d(2,0), - d(0,2), - (d(0,2)*a - d(0,0)),0,0,0,0),(0,0,0, - (d(0,2)*a - 2*d (0,0)),d(2,0),0, - d(0,2)),(0,0,0, - (d(2,0)*a - d(0,2)), - (d(0,2)*a - 2*d(0,0) ),0, - d(2,0)),(0,0,0,0,0, - (d(0,2)*a - 3*d(0,0)),0),(0,0,0, - d(2,0), - d(0,2) ,0,2*d(0,0)))$ $ Unknowns: {d(2,0),d(0,2),d(0,0),a} Unknowns: {d(2,0),d(0,2),d(0,0),a} t3:=D(2,0):= [ 0 1 0 0 0 0 0 ] [ ] [ - a 0 -1 0 0 0 0 ] [ ] [ 1 0 0 0 0 0 0 ] [ ] [ 0 0 0 0 1 0 0 ] [ ] [ 0 0 0 - a 0 0 -1] [ ] [ 0 0 0 0 0 0 0 ] [ ] [ 0 0 0 -1 0 0 0 ] - (a*x + x**3 + 1)*(a*x + x**3 - 1)*x$ {{x,1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [arbcomplex(136)] [ ] [ 0 ] }, 3 {a*x + x - 1, 1, [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ - arbcomplex(137)*x ] [ ] [ 2] [ - arbcomplex(137)*x ] [ ] [ 0 ] [ ] [ arbcomplex(137) ] }, 3 {a*x + x + 1, 1, [arbcomplex(138)*x ] [ ] [ 2] [arbcomplex(138)*x ] [ ] [ arbcomplex(138) ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] [ ] [ 0 ] }} Unknowns: {d(2,0),d(0,2),d(0,0),a} Unknowns: {d(2,0),d(0,2),d(0,0),a} commutant simultane de t1,t2,t3 dans der(gtildedelta): mat((d(0,0),d(2,0),d(0,2),0,0,0,0), ( - (d(2,0)*a - d(0,2)),d(0,0), - d(2,0),0,0,0,0), (d(2,0), - d(0,2), - (d(0,2)*a - d(0,0)),0,0,0,0), (0,0,0, - (d(0,2)*a - 2*d(0,0)),d(2,0),0, - d(0,2)), (0,0,0, - (d(2,0)*a - d(0,2)), - (d(0,2)*a - 2*d(0,0)),0, - d(2,0)), (0,0,0,0,0, - (d(0,2)*a - 3*d(0,0)),0), (0,0,0, - d(2,0), - d(0,2),0,2*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 In the sequel, we suppose that the roots of X**3 +a*X +1 all three distinct. That is 1+4*(a**3)/27 NEQ 0 (2*i*pi)/3 - 3*e and that for a neq{------------------} 2/3 2 P:= [ 1 ] [ 1 ---- l3 0 0 0 0 ] [ l2 ] [ ] [ 2 ] [ l1 1 l3 0 0 0 0 ] [ ] [ 1 1 ] [---- ----- 1 0 0 0 0 ] [ l1 2 ] [ l2 ] [ ] [ - 1 ] [ 0 0 0 1 ------ 0 l3 ] [ l2 ] [ ] [ 2] [ 0 0 0 - l1 1 0 - l3 ] [ ] [ 0 0 0 0 0 1 0 ] [ ] [ 1 - 1 ] [ 0 0 0 ---- ------ 0 1 ] [ l1 2 ] [ l2 ] P**(-1)*t1*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 2 0 0 0] [ ] [0 0 0 0 2 0 0] [ ] [0 0 0 0 0 3 0] [ ] [0 0 0 0 0 0 2] P**(-1)*t2*P:= 3 - ((l1*l2*l3 - 1)*l1 + l2 + l3 + a*l2*l3) - (a*l2 + l2 + 1)*l1*l3 mat((--------------------------------------------,---------------------------, (l1 - l2)*(l1 - l3) 2 (l1 - l2)*(l1 - l3)*l2 3 - (a*l3 + l3 + 1)*l1*l2 ---------------------------,0,0,0,0), (l1 - l2)*(l1 - l3) 3 2 (a*l1 + l1 + 1)*l2 *l3 l1 + l3 + a*l1*l3 + (l1*l2*l3 - 1)*l2 (-------------------------,---------------------------------------, (l1 - l2)*(l2 - l3)*l1 (l1 - l2)*(l2 - l3) 3 2 (a*l3 + l3 + 1)*l1*l2 -------------------------,0,0,0,0), (l1 - l2)*(l2 - l3) 3 3 - (a*l1 + l1 + 1)*l2 - (a*l2 + l2 + 1)*l1 (------------------------,-------------------------, (l1 - l3)*(l2 - l3)*l1 2 (l1 - l3)*(l2 - l3)*l2 - (l1 + l2 + a*l1*l2 + (l1*l2*l3 - 1)*l3) --------------------------------------------,0,0,0,0), (l1 - l3)*(l2 - l3) (l1*l2*l3 - 1)*l1 + l2 + l3 - (l1 - l2 - l3)*a*l1 (0,0,0,---------------------------------------------------, (l1 - l2)*(l1 - l3) 3 3 - (a*l2 + l2 + 1)*l1*l3 (a*l3 + l3 + 1)*l1*l2 ---------------------------,0,------------------------), 2 (l1 - l2)*(l1 - l3) (l1 - l2)*(l1 - l3)*l2 3 2 (a*l1 + l1 + 1)*l2 *l3 (0,0,0,-------------------------, (l1 - l2)*(l2 - l3)*l1 - ((l1*l2*l3 - 1 - a*l2)*l2 + (a*l2 + 1)*(l1 + l3)) ------------------------------------------------------,0, (l1 - l2)*(l2 - l3) 3 2 (a*l3 + l3 + 1)*l1*l2 -------------------------), (l1 - l2)*(l2 - l3) (0,0,0,0,0, - a,0), 3 3 (a*l1 + l1 + 1)*l2 - (a*l2 + l2 + 1)*l1 (0,0,0,------------------------,-------------------------,0, (l1 - l3)*(l2 - l3)*l1 2 (l1 - l3)*(l2 - l3)*l2 (l1*l2*l3 - 1 - a*l3)*l3 + (a*l3 + 1)*(l1 + l2) -------------------------------------------------)) (l1 - l3)*(l2 - l3) P**(-1)*t3*P:= 2 2 3 - (l1 *l2 + l1 *l3 - l1*l2*l3 + 1 + a*l1) - (a*l2 + l2 + 1)*l1 mat((--------------------------------------------,-------------------------, (l1 - l2)*(l1 - l3) 2 (l1 - l2)*(l1 - l3)*l2 3 - (a*l3 + l3 + 1)*l1 ------------------------,0,0,0,0), (l1 - l2)*(l1 - l3) 3 2 (a*l1 + l1 + 1)*l2 ((l1 + l3)*l2 + a)*l2 - (l1*l2*l3 - 1) (------------------------,----------------------------------------, (l1 - l2)*(l2 - l3)*l1 (l1 - l2)*(l2 - l3) 3 2 (a*l3 + l3 + 1)*l2 ----------------------,0,0,0,0), (l1 - l2)*(l2 - l3) 3 3 - (a*l1 + l1 + 1) - (a*l2 + l2 + 1) (------------------------,-------------------------, (l1 - l3)*(l2 - l3)*l1 2 (l1 - l3)*(l2 - l3)*l2 - (((l1 + l2)*l3 + a)*l3 - (l1*l2*l3 - 1)) ---------------------------------------------,0,0,0,0), (l1 - l3)*(l2 - l3) 2 2 3 l1 *l2 + l1 *l3 - l1*l2*l3 + 1 + a*l1 - (a*l2 + l2 + 1)*l1 (0,0,0,---------------------------------------,-------------------------,0, (l1 - l2)*(l1 - l3) 2 (l1 - l2)*(l1 - l3)*l2 3 (a*l3 + l3 + 1)*l1 ---------------------), (l1 - l2)*(l1 - l3) 3 2 (a*l1 + l1 + 1)*l2 (0,0,0,------------------------, (l1 - l2)*(l2 - l3)*l1 3 2 - (((l1 + l3)*l2 + a)*l2 - (l1*l2*l3 - 1)) (a*l3 + l3 + 1)*l2 ---------------------------------------------,0,----------------------), (l1 - l2)*(l2 - l3) (l1 - l2)*(l2 - l3) (0,0,0,0,0,0,0), 3 3 a*l1 + l1 + 1 - (a*l2 + l2 + 1) (0,0,0,------------------------,-------------------------,0, (l1 - l3)*(l2 - l3)*l1 2 (l1 - l3)*(l2 - l3)*l2 ((l1 + l2)*l3 + a)*l3 - (l1*l2*l3 - 1) ----------------------------------------)) (l1 - l3)*(l2 - l3) matrice des derivations dans cette base diagonalisante Y(1),...,Y(7): OK ICI$ PP:= [ 1 ] [ 1 ---- l3 0 0 0 0 ] [ l2 ] [ ] [ 2 ] [ l1 1 l3 0 0 0 0 ] [ ] [ 1 1 ] [---- ----- 1 0 0 0 0 ] [ l1 2 ] [ l2 ] [ ] [ - 1 ] [ 0 0 0 1 ------ 0 l3 ] [ l2 ] [ ] [ 2] [ 0 0 0 - l1 1 0 - l3 ] [ ] [ 0 0 0 0 0 1 0 ] [ ] [ 1 - 1 ] [ 0 0 0 ---- ------ 0 1 ] [ l1 2 ] [ l2 ] avec PP:=P*Q:= [ 1 ] [ 1 ---- l3 0 0 0 0 ] [ l2 ] [ ] [ 2 ] [ l1 1 l3 0 0 0 0 ] [ ] [ 1 1 ] [---- ----- 1 0 0 0 0 ] [ l1 2 ] [ l2 ] [ ] [ - 1 ] [ 0 0 0 1 ------ 0 l3 ] [ l2 ] [ ] [ 2] [ 0 0 0 - l1 1 0 - l3 ] [ ] [ 0 0 0 0 0 1 0 ] [ ] [ 1 - 1 ] [ 0 0 0 ---- ------ 0 1 ] [ l1 2 ] [ l2 ] on voit apparaitre les poids sur la diagonale 2 r(1) := ( - ((d(2,0)*l1 + d(0,2) + d(0,0)*l1)*(l2 + l3) 2 + (d(0,2)*a - d(0,0))*l2*l3 + d(2,0) - d(0,0)*l1 - (d(2,0) - d(0,2)*l1)*l1*l2*l3 + (d(2,0)*a - d(0,2))*l1))/( (l1 - l2)*(l1 - l3)) 2 r(2) := ((d(2,0)*l2 + d(0,2) + d(0,0)*l2)*(l1 + l3) + (d(0,2)*a - d(0,0))*l1*l3 2 + d(2,0) - d(0,0)*l2 - (d(2,0) - d(0,2)*l2)*l1*l2*l3 + (d(2,0)*a - d(0,2))*l2)/((l1 - l2)*(l2 - l3)) 2 r(3) := ( - ((d(2,0)*l3 + d(0,2) + d(0,0)*l3)*(l1 + l2) 2 + (d(0,2)*a - d(0,0))*l1*l2 + d(2,0) - d(0,0)*l3 - (d(2,0) - d(0,2)*l3)*l1*l2*l3 + (d(2,0)*a - d(0,2))*l3))/( (l1 - l3)*(l2 - l3)) 2 r(4) := ((d(0,2)*l1 + 2*d(0,0) - d(2,0)*l1)*l2*l3 2 + (d(0,2)*a - 2*d(0,0))*(l2 + l3)*l1 + (d(2,0)*l1 + d(0,2))*(l2 + l3) 2 + (d(2,0)*a - d(0,2))*l1 - ((d(0,2)*a - 2*d(0,0))*l1 - d(2,0)))/( (l1 - l2)*(l1 - l3)) 2 r(5) := ( - ((d(0,2)*l2 + 2*d(0,0) - d(2,0)*l2)*l1*l3 + (d(0,2)*a - 2*d(0,0))*(l1 + l3)*l2 2 + (d(2,0)*l2 + d(0,2))*(l1 + l3) + (d(2,0)*a - d(0,2))*l2 2 - ((d(0,2)*a - 2*d(0,0))*l2 - d(2,0))))/((l1 - l2)*(l2 - l3)) r(6) := - (d(0,2)*a - 3*d(0,0)) 2 r(7) := ((d(0,2)*l3 + 2*d(0,0) - d(2,0)*l3)*l1*l2 2 + (d(0,2)*a - 2*d(0,0))*(l1 + l2)*l3 + (d(2,0)*l3 + d(0,2))*(l1 + l2) 2 + (d(2,0)*a - d(0,2))*l3 - ((d(0,2)*a - 2*d(0,0))*l3 - d(2,0)))/( (l1 - l3)*(l2 - l3)) r(4)-(r(2)+r(3)):= 0 r(5)-(r(1)+r(3)):= 0 r(7)-(r(1)+r(2)):= 0 r(6)-(r(1)+r(2)+r(3)):= 0 r(1) := gamma1 r(2) := gamma2 r(3) := gamma3 r(4) := - (d(0,2)*a - 3*d(0,0) + gamma1) r(5) := - (d(0,2)*a - 3*d(0,0) + gamma2) r(6) := - (d(0,2)*a - 3*d(0,0)) r(7) := - (d(0,2)*a - 3*d(0,0) + gamma3) Le systeme de poids est le systeme 3.1 calcul de relations de commutation de la base diaY(j) diagonalisant le tore listcommutateursdesx := {{{0,1},x(6)}, {{0,2},x(3)}, {{0,3},a*x(5)}, {{0,4},0}, {{0,5},0}, {{0,6},x(5)}, {{1,2},x(4)}, {{1,3},0}, {{1,4},x(5)}, {{1,5},0}, {{1,6},0}, {{2,3},x(5)}, {{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}} 2 x(2) + x(1)*l1 + x(0)*l1 diaY(1):=--------------------------- l1 2 x(2) + x(1)*l2 + x(0)*l2 diaY(2):=--------------------------- 2 l2 2 diaY(3):=x(2) + x(1)*l3 + x(0)*l3 2 x(6) - x(4)*l1 + x(3)*l1 diaY(4):=--------------------------- l1 2 - x(6) + x(4)*l2 - x(3)*l2 diaY(5):=------------------------------ 2 l2 diaY(6):=x(5) 2 diaY(7):=x(6) - x(4)*l3 + x(3)*l3 liste des commutateurs des diaY(i) :$ The equation X**3 +a*X +1= 0 has distinct roots L1,L2,L3.$ L1,L2,L3 have rather intricate expression which Reduce cannot handle formally.$ Hence in a first step, we keep them as symbols L1,L2,L3.$ listcommutateurdesdiaY:={{{1,2}, (diay(7)*( - l1**4*l2**2 + 2*l1**3*l2**3 - 2*l1**3 - l1**2*l2**4 + 2*l1**2*l2 + 2*l1*l2**2 - 2*l2**3) + diay(5)*l2**2*( - l1**4*l2*l3 + l1**3*l2**2*l3 + l1**3* l2*l3**2 - 2*l1**3 - l1**2*l2**2*l3**2 + l1**2*l2 + l1**2*l3 + l1*l2**2 + l1*l3 **2 - l2**2*l3 - l2*l3**2) + diay(4)*l1*( - l1**2*l2**3*l3 + l1**2*l2**2*l3**2 - l1**2*l2 + l1**2*l3 + l1*l2**4*l3 - l1*l2**3*l3**2 - l1*l2**2 + l1*l3**2 + 2*l2 **3 - l2**2*l3 - l2*l3**2))/(l1*l2**2*(l1**2*l2 - l1**2*l3 - l1*l2**2 + l1*l3**2 + l2**2*l3 - l2*l3**2))}, {{1,3}, (diay(7)*( - l1**4*l2*l3 + l1**3*l2**2*l3 + l1**3*l2*l3**2 - 2*l1**3 - l1**2*l2 **2*l3**2 + l1**2*l2 + l1**2*l3 + l1*l2**2 + l1*l3**2 - l2**2*l3 - l2*l3**2) + diay(5)*l2**2*( - l1**4*l3**2 + 2*l1**3*l3**3 - 2*l1**3 - l1**2*l3**4 + 2*l1**2* l3 + 2*l1*l3**2 - 2*l3**3) + diay(4)*l1*( - l1**2*l2**2*l3**2 + l1**2*l2*l3**3 - l1**2*l2 + l1**2*l3 + l1*l2**2*l3**3 - l1*l2**2 - l1*l2*l3**4 + l1*l3**2 + l2** 2*l3 + l2*l3**2 - 2*l3**3))/(l1*(l1**2*l2 - l1**2*l3 - l1*l2**2 + l1*l3**2 + l2 **2*l3 - l2*l3**2))}, {{1,4},(diay(6)*(a*l1 - l1**3 + 2))/l1}, {{1,5}, (diay(6)*( - a*l1*l2 + l1**2*l2**2 - l1 - l2))/(l1*l2**2)}, {{1,6},0}, {{1,7},(diay(6)*(a*l1*l3 - l1**2*l3**2 + l1 + l3))/l1}, {{2,3}, (diay(7)*( - l1**2*l2**2*l3 - l1**2 + l1*l2**3*l3 - l1*l2 - l1*l3 + 2*l2**2 + l2 *l3) + diay(5)*l2**2*( - l1**2*l2*l3**2 - l1**2 + l1*l2*l3**3 - l1*l2 - l1*l3 + l2*l3 + 2*l3**2) + diay(4)*l1*( - l2**3*l3**2 + l2**2*l3**3 - 2*l2**2 + 2*l3**2) )/(l2**2*(l1**2 - l1*l2 - l1*l3 + l2*l3))}, {{2,4}, (diay(6)*(a*l1*l2 - l1**2*l2**2 + l1 + l2))/(l1*l2**2)}, {{2,5},(diay(6)*( - a*l2 + l2**3 - 2))/l2**3}, {{2,6},0}, {{2,7},(diay(6)*(a*l2*l3 - l2**2*l3**2 + l2 + l3))/l2**2}, {{3,4},(diay(6)*(a*l1*l3 - l1**2*l3**2 + l1 + l3))/l1}, {{3,5},(diay(6)*( - a*l2*l3 + l2**2*l3**2 - l2 - l3))/l2**2}, {{3,6},0}, {{3,7},diay(6)*l3*(a*l3 - l3**3 + 2)}, {{4,5},0}, {{4,6},0}, {{4,7},0}, {{5,6},0}, {{5,7},0}, {{6,7},0}}$ 1 : {1,2}$ C12|7:=( - l1**4*l2**2 + 2*l1**3*l2**3 - 2*l1**3 - l1**2*l2**4 + 2*l1**2*l2 + 2* l1*l2**2 - 2*l2**3)/(l1*l2**2*(l1**2*l2 - l1**2*l3 - l1*l2**2 + l1*l3**2 + l2**2 *l3 - l2*l3**2))$ C12|6:=0$ C12|5:=( - l1**2*l2*l3 - 2*l1 - l2 - l3)/(l1*(l2 - l3))$ C12|4:=( - l1**2*l2**3*l3 + l1**2*l2**2*l3**2 - l1**2*l2 + l1**2*l3 + l1*l2**4* l3 - l1*l2**3*l3**2 - l1*l2**2 + l1*l3**2 + 2*l2**3 - l2**2*l3 - l2*l3**2)/(l2** 2*(l1**2*l2 - l1**2*l3 - l1*l2**2 + l1*l3**2 + l2**2*l3 - l2*l3**2))$ 0$ 2 : {1,3}$ C13|7:=( - l1**2*l2*l3 - 2*l1 - l2 - l3)/(l1*(l2 - l3))$ C13|6:=0$ C13|5:=(l2**2*( - l1**4*l3**2 + 2*l1**3*l3**3 - 2*l1**3 - l1**2*l3**4 + 2*l1**2* l3 + 2*l1*l3**2 - 2*l3**3))/(l1*(l1**2*l2 - l1**2*l3 - l1*l2**2 + l1*l3**2 + l2 **2*l3 - l2*l3**2))$ C13|4:=( - l1**2*l2**2*l3**2 + l1**2*l2*l3**3 - l1**2*l2 + l1**2*l3 + l1*l2**2* l3**3 - l1*l2**2 - l1*l2*l3**4 + l1*l3**2 + l2**2*l3 + l2*l3**2 - 2*l3**3)/(l1** 2*l2 - l1**2*l3 - l1*l2**2 + l1*l3**2 + l2**2*l3 - l2*l3**2)$ 0$ 3 : {1,4}$ C14|7:=0$ C14|6:=(a*l1 - l1**3 + 2)/l1$ C14|5:=0$ C14|4:=0$ 0$ 4 : {1,5}$ C15|7:=0$ C15|6:=( - a*l1*l2 + l1**2*l2**2 - l1 - l2)/(l1*l2**2)$ C15|5:=0$ C15|4:=0$ 5 : {{1,6},0}$ 6 : {1,7}$ C17|7:=0$ C17|6:=(a*l1*l3 - l1**2*l3**2 + l1 + l3)/l1$ C17|5:=0$ C17|4:=0$ 7 :{2,3}$ C23|7:=( - l1**2*l2**2*l3 - l1**2 + l1*l2**3*l3 - l1*l2 - l1*l3 + 2*l2**2 + l2* l3)/(l2**2*(l1**2 - l1*l2 - l1*l3 + l2*l3))$ C23|6:=0$ C23|5:=( - l1**2*l2*l3**2 - l1**2 + l1*l2*l3**3 - l1*l2 - l1*l3 + l2*l3 + 2*l3** 2)/(l1**2 - l1*l2 - l1*l3 + l2*l3)$ C23|4:=(l1*( - l2**3*l3**2 + l2**2*l3**3 - 2*l2**2 + 2*l3**2))/(l2**2*(l1**2 - l1*l2 - l1*l3 + l2*l3))$ 0$ 8 : {2,4}$ C24|7:=0$ C24|6:=(a*l1*l2 - l1**2*l2**2 + l1 + l2)/(l1*l2**2)$ C24|5:=0$ C24|4:=0$ 9 : {2,5}$ C25|7:=0$ C25|6:=( - a*l2 + l2**3 - 2)/l2**3$ C25|5:=0$ C25|4:=0$ 10 : {{2,6},0}$ 11 : {2,7}$ C27|7:=0$ C27|6:=(a*l2*l3 - l2**2*l3**2 + l2 + l3)/l2**2$ C27|5:=0$ C27|4:=0$ 12 : {3,4}$ C34|7:=0$ C34|6:=(a*l1*l3 - l1**2*l3**2 + l1 + l3)/l1$ C34|5:=0$ C34|4:=0$ 13 : {3,5}$ C35|7:=0$ C35|6:=( - a*l2*l3 + l2**2*l3**2 - l2 - l3)/l2**2$ C35|5:=0$ C35|4:=0$ 14 : {{3,6},0}$ 15 : {3,7}$ C37|7:=0$ C37|6:=l3*(a*l3 - l3**3 + 2)$ C37|5:=0$ C37|4:=0$ 16 : {{4,5},0}$ 17 : {{4,6},0}$ 18 : {{4,7},0}$ 19 : {{5,6},0}$ 20 : {{5,7},0}$ 21 : {{6,7},0}$ Now, we record that the following expression vanishes:$ c135*c256 -c146*c234 -c127*c376:= (2*(a*l1**4*l2**2*l3**2 - a*l1**3*l2**3*l3**2 - a*l1**3*l2**2*l3**3 + a*l1**3*l2**2 + a*l1**3*l3**2 + a*l1**2*l2**4*l3**2 - a* l1**2*l2**3*l3**3 + a*l1**2*l2**3 + a*l1**2*l2**2*l3**4 - 2*a*l1**2*l2**2*l3 - 2 *a*l1**2*l2*l3**2 + a*l1**2*l3**3 - 2*a*l1*l2**2*l3**2 + a*l2**3*l3**2 + a*l2**2 *l3**3 - l1**4*l2**4*l3**2 + l1**4*l2**3*l3**3 - l1**4*l2**3 - l1**4*l2**2*l3**4 + 2*l1**4*l2**2*l3 + 2*l1**4*l2*l3**2 - l1**4*l3**3 + l1**3*l2**4*l3**3 - l1**3 *l2**4 + l1**3*l2**3*l3**4 - 2*l1**3*l2**3*l3 - 2*l1**3*l2*l3**3 + 2*l1**3*l2 - l1**3*l3**4 + 2*l1**3*l3 - l1**2*l2**4*l3**4 + 2*l1**2*l2**4*l3 + 2*l1**2*l2*l3 **4 - 4*l1**2*l2*l3 + 2*l1*l2**4*l3**2 - 2*l1*l2**3*l3**3 + 2*l1*l2**3 + 2*l1*l2 **2*l3**4 - 4*l1*l2**2*l3 - 4*l1*l2*l3**2 + 2*l1*l3**3 - l2**4*l3**3 - l2**3*l3 **4 + 2*l2**3*l3 + 2*l2*l3**3))/(l1*l2**2*(l1**2*l2 - l1**2*l3 - l1*l2**2 + l1* l3**2 + l2**2*l3 - l2*l3**2))$ This can be done by a hand calculation$ using the coefficients-roots relations to$ substitute L1*L2*L3 = -1, L1+L2+L3 = 0, L1*L2 + L2*L3 + L3*L1 = a.$ We go back to the formal L1,L2,L3 and we simply for the moment introduce$ the expression of L1**3,L2**3,L4**3 to simplify the 4th and 3rd powers of L1..$ L1**3:=-a*L1 -1$ L2**3:=-a*L2 -1$ L3**3:=-a*L3 -1$ !Then! !Reduces! evaluates! the! preceding! expression! as! zero,! as! we! compu eted! by! ha\ nd! :$ c135*c256 -c146*c234 -c127*c376:= (2*( - 2*a**3*l1**2*l2*l3 - 2*a**3*l1*l2**2*l3 - 2*a**3*l1*l2*l3**2 - 6*a**2*l1**2*l2**2*l3**2 - 4*a**2*l1**2*l2 - 4*a**2*l1** 2*l3 - 4*a**2*l1*l2**2 - 9*a**2*l1*l2*l3 - 4*a**2*l1*l3**2 - 4*a**2*l2**2*l3 - 4 *a**2*l2*l3**2 - 9*a*l1**2*l2**2*l3 - 9*a*l1**2*l2*l3**2 - 6*a*l1**2 - 9*a*l1*l2 **2*l3**2 - 12*a*l1*l2 - 12*a*l1*l3 - 6*a*l2**2 - 12*a*l2*l3 - 6*a*l3**2 - 9*l1 **2*l2*l3 - 9*l1*l2**2*l3 - 9*l1*l2*l3**2 - 9*l1 - 9*l2 - 9*l3))/(a**2*l1*l2 + a *l1**2*l2**2 + a*l1*l2*l3**2 + a*l1 + a*l2 + l1**2*l2**2*l3**2 + l1**2*l2 - l1* l2*l3 + l1*l3**2 + l2**2*l3 + 1)$ The commutation relations read :$ listcommutateurdesdiaY:={{{1,2}, (diay(7)*(2*a**2*l1*l2 + 2*a*l1**2*l2**2 + 4*a*l1 + 4*a*l2 + 3*l1**2*l2 + 3*l1* l2**2 + 6) + diay(5)*( - a**2*l1**2*l2*l3 + a**2*l1*l2**2*l3 + a**2*l1*l2*l3**2 + a*l1**2*l2**2*l3**2 - a*l1**2*l2 - a*l1**2*l3 + a*l1*l2**2 + a*l1*l3**2 + 2*a* l2**2*l3 + 2*a*l2*l3**2 + l1**2*l2**2*l3 + l1**2*l2*l3**2 - l1**2 + l1*l2**2*l3 **2 - l1*l2 - l1*l3 + 2*l2**2 + 2*l2*l3 + 2*l3**2) + diay(4)*( - a**2*l1*l2*l3 - a*l1**2*l2**2*l3 + a*l1**2*l2*l3**2 - a*l1*l2**2*l3**2 - a*l1*l2 - 2*a*l1*l3 - a*l2*l3 - l1**2*l2**2 - l1**2*l2*l3 + 2*l1**2*l3**2 - l1*l2**2*l3 - l1*l2*l3**2 - 2*l1 - l2**2*l3**2 + l2 - 2*l3))/(a**2*l1*l2 + a*l1**2*l2**2 + a*l1*l2*l3**2 + a*l1 + a*l2 + l1**2*l2**2*l3**2 + l1**2*l2 - l1*l2*l3 + l1*l3**2 + l2**2*l3 + 1 )}, {{1,3}, (diay(7)*( - a*l1**2*l2*l3 + a*l1*l2**2*l3 + a*l1*l2*l3**2 - 2*a*l1 + l1**2*l2** 2*l3**2 - l1**2*l2 - l1**2*l3 - l1*l2**2 - l1*l2*l3 - l1*l3**2 + 2*l2**2*l3 + 2* l2*l3**2 - 2) + diay(5)*l2**2*( - 2*a**2*l1*l3 - 2*a*l1**2*l3**2 - 4*a*l1 - 4*a* l3 - 3*l1**2*l3 - 3*l1*l3**2 - 6) + diay(4)*( - a**2*l1*l2*l3 + a*l1**2*l2**2*l3 - a*l1**2*l2*l3**2 - a*l1*l2**2*l3**2 - 2*a*l1*l2 - a*l1*l3 - a*l2*l3 + 2*l1**2 *l2**2 - l1**2*l2*l3 - l1**2*l3**2 - l1*l2**2*l3 - l1*l2*l3**2 - 2*l1 - l2**2*l3 **2 - 2*l2 + l3))/(a*l1*l2 - a*l1*l3 + l1**2*l2**2 - l1**2*l3**2 - l1*l2**2*l3 + l1*l2*l3**2 + l2 - l3)}, {{1,4},(diay(6)*(2*a*l1 + 3))/l1}, {{1,5}, (diay(6)*( - a*l1*l2 + l1**2*l2**2 - l1 - l2))/(l1*l2**2)}, {{1,6},0}, {{1,7},(diay(6)*(a*l1*l3 - l1**2*l3**2 + l1 + l3))/l1}, {{2,3}, (diay(7)*( - a*l1*l2*l3 - l1**2*l2**2*l3 - l1**2 - l1*l2 - 2*l1*l3 + 2*l2**2 + l2*l3) + diay(5)*(a**2*l1*l2*l3 + a*l1**2*l2*l3**2 + 2*a*l1*l2 + a*l1*l3 - a*l2* l3 - l1**2*l2**2 + l1**2*l3**2 - l1*l2**2*l3 + 2*l1 + 2*l2**2*l3**2 - l3) + diay (4)*l1*( - a*l2**2*l3 + a*l2*l3**2 - 3*l2**2 + 3*l3**2))/(a*l1*l2 - a*l2*l3 + l1 **2*l2**2 - l1*l2**2*l3 + l1 - l3)}, {{2,4}, (diay(6)*(a*l1*l2 - l1**2*l2**2 + l1 + l2))/(l1*l2**2)}, {{2,5},(diay(6)*(2*a*l2 + 3))/(a*l2 + 1)}, {{2,6},0}, {{2,7},(diay(6)*(a*l2*l3 - l2**2*l3**2 + l2 + l3))/l2**2}, {{3,4},(diay(6)*(a*l1*l3 - l1**2*l3**2 + l1 + l3))/l1}, {{3,5},(diay(6)*( - a*l2*l3 + l2**2*l3**2 - l2 - l3))/l2**2}, {{3,6},0}, {{3,7},diay(6)*l3*(2*a*l3 + 3)}, {{4,5},0}, {{4,6},0}, {{4,7},0}, {{5,6},0}, {{5,7},0}, {{6,7},0}}$ 1 : {1,2}$ C12|7:=(2*a**2*l1*l2 + 2*a*l1**2*l2**2 + 4*a*l1 + 4*a*l2 + 3*l1**2*l2 + 3*l1*l2 **2 + 6)/(a**2*l1*l2 + a*l1**2*l2**2 + a*l1*l2*l3**2 + a*l1 + a*l2 + l1**2*l2**2 *l3**2 + l1**2*l2 - l1*l2*l3 + l1*l3**2 + l2**2*l3 + 1)$ C12|6:=0$ C12|5:=( - a**2*l1**2*l2*l3 + a**2*l1*l2**2*l3 + a**2*l1*l2*l3**2 + a*l1**2*l2** 2*l3**2 - a*l1**2*l2 - a*l1**2*l3 + a*l1*l2**2 + a*l1*l3**2 + 2*a*l2**2*l3 + 2*a *l2*l3**2 + l1**2*l2**2*l3 + l1**2*l2*l3**2 - l1**2 + l1*l2**2*l3**2 - l1*l2 - l1*l3 + 2*l2**2 + 2*l2*l3 + 2*l3**2)/(a**2*l1*l2 + a*l1**2*l2**2 + a*l1*l2*l3**2 + a*l1 + a*l2 + l1**2*l2**2*l3**2 + l1**2*l2 - l1*l2*l3 + l1*l3**2 + l2**2*l3 + 1)$ C12|4:=( - a**2*l1*l2*l3 - a*l1**2*l2**2*l3 + a*l1**2*l2*l3**2 - a*l1*l2**2*l3** 2 - a*l1*l2 - 2*a*l1*l3 - a*l2*l3 - l1**2*l2**2 - l1**2*l2*l3 + 2*l1**2*l3**2 - l1*l2**2*l3 - l1*l2*l3**2 - 2*l1 - l2**2*l3**2 + l2 - 2*l3)/(a**2*l1*l2 + a*l1** 2*l2**2 + a*l1*l2*l3**2 + a*l1 + a*l2 + l1**2*l2**2*l3**2 + l1**2*l2 - l1*l2*l3 + l1*l3**2 + l2**2*l3 + 1)$ 0$ 2 : {1,3}$ C13|7:=( - a*l1**2*l2*l3 + a*l1*l2**2*l3 + a*l1*l2*l3**2 - 2*a*l1 + l1**2*l2**2* l3**2 - l1**2*l2 - l1**2*l3 - l1*l2**2 - l1*l2*l3 - l1*l3**2 + 2*l2**2*l3 + 2*l2 *l3**2 - 2)/(a*l1*l2 - a*l1*l3 + l1**2*l2**2 - l1**2*l3**2 - l1*l2**2*l3 + l1*l2 *l3**2 + l2 - l3)$ C13|6:=0$ C13|5:=(l2**2*( - 2*a**2*l1*l3 - 2*a*l1**2*l3**2 - 4*a*l1 - 4*a*l3 - 3*l1**2*l3 - 3*l1*l3**2 - 6))/(a*l1*l2 - a*l1*l3 + l1**2*l2**2 - l1**2*l3**2 - l1*l2**2*l3 + l1*l2*l3**2 + l2 - l3)$ C13|4:=( - a**2*l1*l2*l3 + a*l1**2*l2**2*l3 - a*l1**2*l2*l3**2 - a*l1*l2**2*l3** 2 - 2*a*l1*l2 - a*l1*l3 - a*l2*l3 + 2*l1**2*l2**2 - l1**2*l2*l3 - l1**2*l3**2 - l1*l2**2*l3 - l1*l2*l3**2 - 2*l1 - l2**2*l3**2 - 2*l2 + l3)/(a*l1*l2 - a*l1*l3 + l1**2*l2**2 - l1**2*l3**2 - l1*l2**2*l3 + l1*l2*l3**2 + l2 - l3)$ 0$ 3 : {1,4}$ C14|7:=0$ C14|6:=(2*a*l1 + 3)/l1$ C14|5:=0$ C14|4:=0$ 0$ 4 : {1,5}$ C15|7:=0$ C15|6:=( - a*l1*l2 + l1**2*l2**2 - l1 - l2)/(l1*l2**2)$ C15|5:=0$ C15|4:=0$ 5 : {{1,6},0}$ 6 : {1,7}$ C17|7:=0$ C17|6:=(a*l1*l3 - l1**2*l3**2 + l1 + l3)/l1$ C17|5:=0$ C17|4:=0$ 7 :{2,3}$ C23|7:=( - a*l1*l2*l3 - l1**2*l2**2*l3 - l1**2 - l1*l2 - 2*l1*l3 + 2*l2**2 + l2* l3)/(a*l1*l2 - a*l2*l3 + l1**2*l2**2 - l1*l2**2*l3 + l1 - l3)$ C23|6:=0$ C23|5:=(a**2*l1*l2*l3 + a*l1**2*l2*l3**2 + 2*a*l1*l2 + a*l1*l3 - a*l2*l3 - l1**2 *l2**2 + l1**2*l3**2 - l1*l2**2*l3 + 2*l1 + 2*l2**2*l3**2 - l3)/(a*l1*l2 - a*l2* l3 + l1**2*l2**2 - l1*l2**2*l3 + l1 - l3)$ C23|4:=(l1*( - a*l2**2*l3 + a*l2*l3**2 - 3*l2**2 + 3*l3**2))/(a*l1*l2 - a*l2*l3 + l1**2*l2**2 - l1*l2**2*l3 + l1 - l3)$ 0$ 8 : {2,4}$ C24|7:=0$ C24|6:=(a*l1*l2 - l1**2*l2**2 + l1 + l2)/(l1*l2**2)$ C24|5:=0$ C24|4:=0$ 9 : {2,5}$ C25|7:=0$ C25|6:=(2*a*l2 + 3)/(a*l2 + 1)$ C25|5:=0$ C25|4:=0$ 10 : {{2,6},0}$ 11 : {2,7}$ C27|7:=0$ C27|6:=(a*l2*l3 - l2**2*l3**2 + l2 + l3)/l2**2$ C27|5:=0$ C27|4:=0$ 12 : {3,4}$ C34|7:=0$ C34|6:=(a*l1*l3 - l1**2*l3**2 + l1 + l3)/l1$ C34|5:=0$ C34|4:=0$ 13 : {3,5}$ C35|7:=0$ C35|6:=( - a*l2*l3 + l2**2*l3**2 - l2 - l3)/l2**2$ C35|5:=0$ C35|4:=0$ 14 : {{3,6},0}$ 15 : {3,7}$ C37|7:=0$ C37|6:=l3*(2*a*l3 + 3)$ C37|5:=0$ C37|4:=0$ 16 : {{4,5},0}$ 17 : {{4,6},0}$ 18 : {{4,7},0}$ 19 : {{5,6},0}$ 20 : {{5,7},0}$ 21 : {{6,7},0}$ In the second step, we will use the coefficients-roots relations to$ substitute L1*L2*L3 = -1, L1+L2+L3 = 0, L1*L2 + L2*L3 + L3*L1 = a.$ Then we see by a hand calculation that,$ in full accordance with the root pattern of the Lie algebra,$ only C127, C135, C146, C234, C256, C376 are nonzero.$ This can be checked in the case a=3 with the procedure avalue3 :$ We take a:=3$ listcommutateurdesdiaY:={{{1,2},(1.9832780319 - 0.18211071332*i)*diay(7)}, {{1,3},(0.1557051041 + 1.8130543645*i)*diay(5)}, {{1,4},(6.1557051041 - 1.695707555*i)*diay(6)}, {{1,5},0}, {{1,6},0}, {{1,7},0}, {{2,3},( - 0.050165904183 + 0.58413956339*i)*diay(4)}, {{2,4},0}, {{2,5},(2.0496066558 + 0.17602021426*i)*diay(6)}, {{2,6},0}, {{2,7},0}, {{3,4},0}, {{3,5},0}, {{3,6},0}, {{3,7}, - 0.34373564747*diay(6)}, {{4,5},0}, {{4,6},0}, {{4,7},0}, {{5,6},0}, {{5,7},0}, {{6,7},0}}$ 1 : {1,2}$ C12|7:=1.9832780319 - 0.18211071332*i$ C12|6:=0$ C12|5:=0$ C12|4:=0$ 0$ 2 : {1,3}$ C13|7:=0$ C13|6:=0$ C13|5:=0.1557051041 + 1.8130543645*i$ C13|4:=0$ 0$ 3 : {1,4}$ C14|7:=0$ C14|6:=6.1557051041 - 1.695707555*i$ C14|5:=0$ C14|4:=0$ 0$ 4 : {1,5}$ C15|7:=0$ C15|6:=0$ C15|5:=0$ C15|4:=0$ 5 : {{1,6},0}$ 6 : {1,7}$ C17|7:=0$ C17|6:=0$ C17|5:=0$ C17|4:=0$ 7 :{2,3}$ C23|7:=0$ C23|6:=0$ C23|5:=0$ C23|4:= - 0.050165904183 + 0.58413956339*i$ 0$ 8 : {2,4}$ C24|7:=0$ C24|6:=0$ C24|5:=0$ C24|4:=0$ 9 : {2,5}$ C25|7:=0$ C25|6:=2.0496066558 + 0.17602021426*i$ C25|5:=0$ C25|4:=0$ 10 : {{2,6},0}$ 11 : {2,7}$ C27|7:=0$ C27|6:=0$ C27|5:=0$ C27|4:=0$ 12 : {3,4}$ C34|7:=0$ C34|6:=0$ C34|5:=0$ C34|4:=0$ 13 : {3,5}$ C35|7:=0$ C35|6:=0$ C35|5:=0$ C35|4:=0$ 14 : {{3,6},0}$ 15 : {3,7}$ C37|7:=0$ C37|6:= - 0.34373564747$ C37|5:=0$ C37|4:=0$ 16 : {{4,5},0}$ 17 : {{4,6},0}$ 18 : {{4,7},0}$ 19 : {{5,6},0}$ 20 : {{5,7},0}$ 21 : {{6,7},0}$ L:= l$ c135*c256/(c146*c234):= 0.98327803194 + 0.18211071332*i$ We return to the general case for a.$ Return to general case for a.$ listcommutateurdesdiaY:={{{1,2}, (diay(7)*(2*a**2*l1*l2 + 2*a*l1**2*l2**2 + 4*a*l1 + 4*a*l2 + 3*l1**2*l2 + 3*l1* l2**2 + 6) + diay(5)*( - a**2*l1**2*l2*l3 + a**2*l1*l2**2*l3 + a**2*l1*l2*l3**2 + a*l1**2*l2**2*l3**2 - a*l1**2*l2 - a*l1**2*l3 + a*l1*l2**2 + a*l1*l3**2 + 2*a* l2**2*l3 + 2*a*l2*l3**2 + l1**2*l2**2*l3 + l1**2*l2*l3**2 - l1**2 + l1*l2**2*l3 **2 - l1*l2 - l1*l3 + 2*l2**2 + 2*l2*l3 + 2*l3**2) + diay(4)*( - a**2*l1*l2*l3 - a*l1**2*l2**2*l3 + a*l1**2*l2*l3**2 - a*l1*l2**2*l3**2 - a*l1*l2 - 2*a*l1*l3 - a*l2*l3 - l1**2*l2**2 - l1**2*l2*l3 + 2*l1**2*l3**2 - l1*l2**2*l3 - l1*l2*l3**2 - 2*l1 - l2**2*l3**2 + l2 - 2*l3))/(a**2*l1*l2 + a*l1**2*l2**2 + a*l1*l2*l3**2 + a*l1 + a*l2 + l1**2*l2**2*l3**2 + l1**2*l2 - l1*l2*l3 + l1*l3**2 + l2**2*l3 + 1 )}, {{1,3}, (diay(7)*( - a*l1**2*l2*l3 + a*l1*l2**2*l3 + a*l1*l2*l3**2 - 2*a*l1 + l1**2*l2** 2*l3**2 - l1**2*l2 - l1**2*l3 - l1*l2**2 - l1*l2*l3 - l1*l3**2 + 2*l2**2*l3 + 2* l2*l3**2 - 2) + diay(5)*l2**2*( - 2*a**2*l1*l3 - 2*a*l1**2*l3**2 - 4*a*l1 - 4*a* l3 - 3*l1**2*l3 - 3*l1*l3**2 - 6) + diay(4)*( - a**2*l1*l2*l3 + a*l1**2*l2**2*l3 - a*l1**2*l2*l3**2 - a*l1*l2**2*l3**2 - 2*a*l1*l2 - a*l1*l3 - a*l2*l3 + 2*l1**2 *l2**2 - l1**2*l2*l3 - l1**2*l3**2 - l1*l2**2*l3 - l1*l2*l3**2 - 2*l1 - l2**2*l3 **2 - 2*l2 + l3))/(a*l1*l2 - a*l1*l3 + l1**2*l2**2 - l1**2*l3**2 - l1*l2**2*l3 + l1*l2*l3**2 + l2 - l3)}, {{1,4},(diay(6)*(2*a*l1 + 3))/l1}, {{1,5}, (diay(6)*( - a*l1*l2 + l1**2*l2**2 - l1 - l2))/(l1*l2**2)}, {{1,6},0}, {{1,7},(diay(6)*(a*l1*l3 - l1**2*l3**2 + l1 + l3))/l1}, {{2,3}, (diay(7)*( - a*l1*l2*l3 - l1**2*l2**2*l3 - l1**2 - l1*l2 - 2*l1*l3 + 2*l2**2 + l2*l3) + diay(5)*(a**2*l1*l2*l3 + a*l1**2*l2*l3**2 + 2*a*l1*l2 + a*l1*l3 - a*l2* l3 - l1**2*l2**2 + l1**2*l3**2 - l1*l2**2*l3 + 2*l1 + 2*l2**2*l3**2 - l3) + diay (4)*l1*( - a*l2**2*l3 + a*l2*l3**2 - 3*l2**2 + 3*l3**2))/(a*l1*l2 - a*l2*l3 + l1 **2*l2**2 - l1*l2**2*l3 + l1 - l3)}, {{2,4}, (diay(6)*(a*l1*l2 - l1**2*l2**2 + l1 + l2))/(l1*l2**2)}, {{2,5},(diay(6)*(2*a*l2 + 3))/(a*l2 + 1)}, {{2,6},0}, {{2,7},(diay(6)*(a*l2*l3 - l2**2*l3**2 + l2 + l3))/l2**2}, {{3,4},(diay(6)*(a*l1*l3 - l1**2*l3**2 + l1 + l3))/l1}, {{3,5},(diay(6)*( - a*l2*l3 + l2**2*l3**2 - l2 - l3))/l2**2}, {{3,6},0}, {{3,7},diay(6)*l3*(2*a*l3 + 3)}, {{4,5},0}, {{4,6},0}, {{4,7},0}, {{5,6},0}, {{5,7},0}, {{6,7},0}}$ 1 : {1,2}$ C12|7:=(2*a**2*l1*l2 + 2*a*l1**2*l2**2 + 4*a*l1 + 4*a*l2 + 3*l1**2*l2 + 3*l1*l2 **2 + 6)/(a**2*l1*l2 + a*l1**2*l2**2 + a*l1*l2*l3**2 + a*l1 + a*l2 + l1**2*l2**2 *l3**2 + l1**2*l2 - l1*l2*l3 + l1*l3**2 + l2**2*l3 + 1)$ C12|6:=0$ C12|5:=( - a**2*l1**2*l2*l3 + a**2*l1*l2**2*l3 + a**2*l1*l2*l3**2 + a*l1**2*l2** 2*l3**2 - a*l1**2*l2 - a*l1**2*l3 + a*l1*l2**2 + a*l1*l3**2 + 2*a*l2**2*l3 + 2*a *l2*l3**2 + l1**2*l2**2*l3 + l1**2*l2*l3**2 - l1**2 + l1*l2**2*l3**2 - l1*l2 - l1*l3 + 2*l2**2 + 2*l2*l3 + 2*l3**2)/(a**2*l1*l2 + a*l1**2*l2**2 + a*l1*l2*l3**2 + a*l1 + a*l2 + l1**2*l2**2*l3**2 + l1**2*l2 - l1*l2*l3 + l1*l3**2 + l2**2*l3 + 1)$ C12|4:=( - a**2*l1*l2*l3 - a*l1**2*l2**2*l3 + a*l1**2*l2*l3**2 - a*l1*l2**2*l3** 2 - a*l1*l2 - 2*a*l1*l3 - a*l2*l3 - l1**2*l2**2 - l1**2*l2*l3 + 2*l1**2*l3**2 - l1*l2**2*l3 - l1*l2*l3**2 - 2*l1 - l2**2*l3**2 + l2 - 2*l3)/(a**2*l1*l2 + a*l1** 2*l2**2 + a*l1*l2*l3**2 + a*l1 + a*l2 + l1**2*l2**2*l3**2 + l1**2*l2 - l1*l2*l3 + l1*l3**2 + l2**2*l3 + 1)$ 0$ 2 : {1,3}$ C13|7:=( - a*l1**2*l2*l3 + a*l1*l2**2*l3 + a*l1*l2*l3**2 - 2*a*l1 + l1**2*l2**2* l3**2 - l1**2*l2 - l1**2*l3 - l1*l2**2 - l1*l2*l3 - l1*l3**2 + 2*l2**2*l3 + 2*l2 *l3**2 - 2)/(a*l1*l2 - a*l1*l3 + l1**2*l2**2 - l1**2*l3**2 - l1*l2**2*l3 + l1*l2 *l3**2 + l2 - l3)$ C13|6:=0$ C13|5:=(l2**2*( - 2*a**2*l1*l3 - 2*a*l1**2*l3**2 - 4*a*l1 - 4*a*l3 - 3*l1**2*l3 - 3*l1*l3**2 - 6))/(a*l1*l2 - a*l1*l3 + l1**2*l2**2 - l1**2*l3**2 - l1*l2**2*l3 + l1*l2*l3**2 + l2 - l3)$ C13|4:=( - a**2*l1*l2*l3 + a*l1**2*l2**2*l3 - a*l1**2*l2*l3**2 - a*l1*l2**2*l3** 2 - 2*a*l1*l2 - a*l1*l3 - a*l2*l3 + 2*l1**2*l2**2 - l1**2*l2*l3 - l1**2*l3**2 - l1*l2**2*l3 - l1*l2*l3**2 - 2*l1 - l2**2*l3**2 - 2*l2 + l3)/(a*l1*l2 - a*l1*l3 + l1**2*l2**2 - l1**2*l3**2 - l1*l2**2*l3 + l1*l2*l3**2 + l2 - l3)$ 0$ 3 : {1,4}$ C14|7:=0$ C14|6:=(2*a*l1 + 3)/l1$ C14|5:=0$ C14|4:=0$ 0$ 4 : {1,5}$ C15|7:=0$ C15|6:=( - a*l1*l2 + l1**2*l2**2 - l1 - l2)/(l1*l2**2)$ C15|5:=0$ C15|4:=0$ 5 : {{1,6},0}$ 6 : {1,7}$ C17|7:=0$ C17|6:=(a*l1*l3 - l1**2*l3**2 + l1 + l3)/l1$ C17|5:=0$ C17|4:=0$ 7 :{2,3}$ C23|7:=( - a*l1*l2*l3 - l1**2*l2**2*l3 - l1**2 - l1*l2 - 2*l1*l3 + 2*l2**2 + l2* l3)/(a*l1*l2 - a*l2*l3 + l1**2*l2**2 - l1*l2**2*l3 + l1 - l3)$ C23|6:=0$ C23|5:=(a**2*l1*l2*l3 + a*l1**2*l2*l3**2 + 2*a*l1*l2 + a*l1*l3 - a*l2*l3 - l1**2 *l2**2 + l1**2*l3**2 - l1*l2**2*l3 + 2*l1 + 2*l2**2*l3**2 - l3)/(a*l1*l2 - a*l2* l3 + l1**2*l2**2 - l1*l2**2*l3 + l1 - l3)$ C23|4:=(l1*( - a*l2**2*l3 + a*l2*l3**2 - 3*l2**2 + 3*l3**2))/(a*l1*l2 - a*l2*l3 + l1**2*l2**2 - l1*l2**2*l3 + l1 - l3)$ 0$ 8 : {2,4}$ C24|7:=0$ C24|6:=(a*l1*l2 - l1**2*l2**2 + l1 + l2)/(l1*l2**2)$ C24|5:=0$ C24|4:=0$ 9 : {2,5}$ C25|7:=0$ C25|6:=(2*a*l2 + 3)/(a*l2 + 1)$ C25|5:=0$ C25|4:=0$ 10 : {{2,6},0}$ 11 : {2,7}$ C27|7:=0$ C27|6:=(a*l2*l3 - l2**2*l3**2 + l2 + l3)/l2**2$ C27|5:=0$ C27|4:=0$ 12 : {3,4}$ C34|7:=0$ C34|6:=(a*l1*l3 - l1**2*l3**2 + l1 + l3)/l1$ C34|5:=0$ C34|4:=0$ 13 : {3,5}$ C35|7:=0$ C35|6:=( - a*l2*l3 + l2**2*l3**2 - l2 - l3)/l2**2$ C35|5:=0$ C35|4:=0$ 14 : {{3,6},0}$ 15 : {3,7}$ C37|7:=0$ C37|6:=l3*(2*a*l3 + 3)$ C37|5:=0$ C37|4:=0$ 16 : {{4,5},0}$ 17 : {{4,6},0}$ 18 : {{4,7},0}$ 19 : {{5,6},0}$ 20 : {{5,7},0}$ 21 : {{6,7},0}$ Now we set :$ Z(1):=diaY(1)$ Z(2):=diaY(2)$ Z(3):=diaY(3)$ Z(4):=C127*diaY(7)$ Z(5):=C135*diaY(5)$ Z(6):=C234*diaY(4)$ Z(7):=C146*C234*diaY(6)$ L:=(C135*C256)/(C146*C234)$ Then we get the commutation relations :$ [Z(1),Z(2)]:=Z(4)$ [Z(1),Z(3)]:=Z(5)$ [Z(2),Z(3)]:=Z(6)$ [Z(1),Z(6)]:=Z(7)$ [Z(2),Z(5)]:=L*Z(7)$ and [Z(2),Z(5)]:=((C127*C376)/(C146*C234))*Z(7)$ Now a reduce computation shows that :$ c135*c256 -c146*c234 -c127*c376:= 0$ Hence the last relation reads$ and [Z(2),Z(5)]:=(L-1)*Z(7)$ This simply means that the Jacobi Identity for Z(1),Z(2),Z(3) is verified,$ as it must be since gtildedelta IS a Lie algebra!$ We get the commutation relations of$ g_{7,3.1}$ (iL)$ with L:=$ L:=(C135*C256)/(C146*C234)$ C135:= (l2**2*( - 2*a**2*l1*l3 - 2*a*l1**2*l3**2 - 4*a*l1 - 4*a*l3 - 3*l1**2*l3 - 3*l1*l3**2 - 6))/(a*l1*l2 - a*l1*l3 + l1**2*l2**2 - l1**2*l3**2 - l1*l2**2*l3 + l1*l2*l3**2 + l2 - l3)$ C256:= (2*a*l2 + 3)/(a*l2 + 1)$ C146:= (2*a*l1 + 3)/l1$ C234:= ( - (3*(l2 + l3) + a*l2*l3)*(l2 - l3)*l1)/((l1*l2**2 + 1 + a*l2)*(l1 - l3 ))$ L:=((4*a**5*l1*l2 + 4*a**4*l1**2*l2**2 + 4*a**4*l1**2*l2*l3 - 4*a**4*l1*l2**2*l3 + 4*a**4*l1 + 4*a**4*l2 + 8*a**3*l1**2*l2 + 4*a**3*l1**2*l3 - 4*a**3*l1*l2**2 - 8*a**3*l1*l2*l3 - 6*a**3*l2**2*l3 + 4*a**3 - 10*a**2*l1**2*l2**2*l3 + 4*a**2*l1 **2 + 15*a**2*l1*l2 - 4*a**2*l1*l3 - 10*a**2*l2**2 - 21*a**2*l2*l3 + 9*a*l1**2* l2**2 - 6*a*l1**2*l2*l3 - 9*a*l1*l2**2*l3 + 15*a*l1 - 6*a*l2 - 15*a*l3 + 9*l1**2 *l2 - 9*l1*l2**2 - 9*l1*l2*l3 + 9*l2**2*l3)*(2*a*l3 + 3))/((4*a**4*l1*l2 + 4*a** 3*l1**2*l2**2 + 4*a**3*l1*l2*l3**2 + 4*a**3*l1 + 4*a**3*l2 + 4*a**2*l1**2*l2**2* l3**2 + 4*a**2*l1*l2**2 - 4*a**2*l1*l2*l3 + 4*a**2*l1*l3**2 + 2*a**2*l2**2*l3 + 2*a**2*l2*l3**2 + 4*a**2 - 2*a*l1**2*l2**2*l3 - 2*a*l1**2*l2*l3**2 - 4*a*l1**2 + 4*a*l1*l2**2*l3**2 + 23*a*l1*l2 - 4*a*l1*l3 + 2*a*l2**2 - a*l2*l3 + 2*a*l3**2 + 21*l1**2*l2**2 + 3*l1**2*l2*l3 - 6*l1**2*l3**2 - 6*l1*l2**2*l3 + 21*l1*l2*l3**2 + 15*l1 + 3*(l2**2*l3**2 + 8*l2 - l3))*a + 9*((2*l1*l2**2*l3**2 + l1*l2 + l1*l3 + l2**2 - 2*l2*l3 + l3**2)*l1 + l2**2*l3 + l2*l3**2 + 2))$ Et cela pour a:=a$ and that for a neq {( - 3*e**((2*i*pi)/3))/2**(2/3)}$ shortformdelta:={0, ss, 0, 1, ss, a, 1, ss, 1, 0, 0}$ delta:= mat((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,a,0,0,1),(1,0,0 ,0,0,0))$ $ We check the value of L as written here on the avalue a=3:$ !L:=-(9*l2**2! +2*a**2*l2! +6*a)*l1*l2**2*(2*a*l2+3)/((! 2*a*l1+3)**2*(l2-l3)**2 *(\ a*l2+1))$ We take a:=3$ listcommutateurdesdiaY:={{{1,2},(1.9832780319 - 0.18211071332*i)*diay(7)}, {{1,3},(0.1557051041 + 1.8130543645*i)*diay(5)}, {{1,4},(6.1557051041 - 1.695707555*i)*diay(6)}, {{1,5},0}, {{1,6},0}, {{1,7},0}, {{2,3},( - 0.050165904183 + 0.58413956339*i)*diay(4)}, {{2,4},0}, {{2,5},(2.0496066558 + 0.17602021426*i)*diay(6)}, {{2,6},0}, {{2,7},0}, {{3,4},0}, {{3,5},0}, {{3,6},0}, {{3,7}, - 0.34373564747*diay(6)}, {{4,5},0}, {{4,6},0}, {{4,7},0}, {{5,6},0}, {{5,7},0}, {{6,7},0}}$ 1 : {1,2}$ C12|7:=1.9832780319 - 0.18211071332*i$ C12|6:=0$ C12|5:=0$ C12|4:=0$ 0$ 2 : {1,3}$ C13|7:=0$ C13|6:=0$ C13|5:=0.1557051041 + 1.8130543645*i$ C13|4:=0$ 0$ 3 : {1,4}$ C14|7:=0$ C14|6:=6.1557051041 - 1.695707555*i$ C14|5:=0$ C14|4:=0$ 0$ 4 : {1,5}$ C15|7:=0$ C15|6:=0$ C15|5:=0$ C15|4:=0$ 5 : {{1,6},0}$ 6 : {1,7}$ C17|7:=0$ C17|6:=0$ C17|5:=0$ C17|4:=0$ 7 :{2,3}$ C23|7:=0$ C23|6:=0$ C23|5:=0$ C23|4:= - 0.050165904183 + 0.58413956339*i$ 0$ 8 : {2,4}$ C24|7:=0$ C24|6:=0$ C24|5:=0$ C24|4:=0$ 9 : {2,5}$ C25|7:=0$ C25|6:=2.0496066558 + 0.17602021426*i$ C25|5:=0$ C25|4:=0$ 10 : {{2,6},0}$ 11 : {2,7}$ C27|7:=0$ C27|6:=0$ C27|5:=0$ C27|4:=0$ 12 : {3,4}$ C34|7:=0$ C34|6:=0$ C34|5:=0$ C34|4:=0$ 13 : {3,5}$ C35|7:=0$ C35|6:=0$ C35|5:=0$ C35|4:=0$ 14 : {{3,6},0}$ 15 : {3,7}$ C37|7:=0$ C37|6:= - 0.34373564747$ C37|5:=0$ C37|4:=0$ 16 : {{4,5},0}$ 17 : {{4,6},0}$ 18 : {{4,7},0}$ 19 : {{5,6},0}$ 20 : {{5,7},0}$ 21 : {{6,7},0}$ L:= 0.98327803194 + 0.18211071332*i$ c135*c256/(c146*c234):= 0.98327803194 + 0.18211071332*i$ We return to the general case for a.$