The generic automorphism phi of g_{6,2} as computed by calculautom6_2.red : phi:= mat((b(1,1),0,0,0,0,0), b(4,4)*b(3,3) - b(4,3)*b(3,4) (b(2,1),-------------------------------,0,0,0,0), 2 b(1,1) (b(3,1),0,b(3,3),b(3,4),0,0), (b(4,1),0,b(4,3),b(4,4),0,0), - b(4,3)*b(3,1) + b(4,1)*b(3,3) (b(5,1),b(5,2),----------------------------------, b(1,1) - b(4,4)*b(3,1) + b(4,1)*b(3,4) b(4,4)*b(3,3) - b(4,3)*b(3,4) ----------------------------------,-------------------------------,0), b(1,1) b(1,1) (b(6,1),b(6,2),b(6,3),b(6,4),b(5,2)*b(1,1),b(4,4)*b(3,3) - b(4,3)*b(3,4))) 4 (b(4,4)*b(3,3) - b(4,3)*b(3,4)) det(phi):=---------------------------------- 2 b(1,1) 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),0,xi(3,3),xi(3,4),0,0), (xi(4,1),0,xi(4,3),xi(2,2) + 2*xi(1,1) - xi(3,3),0,0), (xi(5,1),xi(5,2),xi(4,1), - xi(3,1),xi(2,2) + xi(1,1),0), (xi(6,1),xi(6,2),xi(6,3),xi(6,4),xi(5,2),xi(2,2) + 2*xi(1,1))) [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] adx1 := [ ] [0 0 0 0 0 0] [ ] [0 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 0 0 0 0] adx2 := [ ] [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 0] adx3 := [ ] [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 0 0 0 0 0] adx4 := [ ] [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] [ ] [0 0 0 0 0 0] adx5 := [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [-1 0 0 0 0 0] The generic nilpotent derivation : the eigenvalues are 0 xi(1,1):=0 xi(2,2):=0 and in case 2 one has xi(3,3):=0 xi(3,4):=0 xi(4,3):=0 by subtracting adjoints one then may suppose: xi(5,1):=0,xi(5,2):=0,xi(6,1):=0,xi(6,3):=0,xi(6,4):=0 phi:= mat((b(1,1),0,0,0,0,0), b(4,4)*b(3,3) - b(4,3)*b(3,4) (b(2,1),-------------------------------,0,0,0,0), 2 b(1,1) (b(3,1),0,b(3,3),b(3,4),0,0), (b(4,1),0,b(4,3),b(4,4),0,0), - (b(4,3)*b(3,1) - b(4,1)*b(3,3)) (b(5,1),b(5,2),------------------------------------, b(1,1) - (b(4,4)*b(3,1) - b(4,1)*b(3,4)) b(4,4)*b(3,3) - b(4,3)*b(3,4) ------------------------------------,-------------------------------,0), b(1,1) b(1,1) (b(6,1),b(6,2),b(6,3),b(6,4),b(5,2)*b(1,1),b(4,4)*b(3,3) - b(4,3)*b(3,4))) 4 (b(4,4)*b(3,3) - b(4,3)*b(3,4)) det(phi):=---------------------------------- 2 b(1,1) delta:= [ 0 0 0 0 0 0] [ ] [xi(2,1) 0 0 0 0 0] [ ] [xi(3,1) 0 0 0 0 0] [ ] [xi(4,1) 0 0 0 0 0] [ ] [ 0 0 xi(4,1) - xi(3,1) 0 0] [ ] [ 0 xi(6,2) 0 0 0 0] We denote this delta by the shortform shortformdelta:={xi(2,1), ss, xi(3,1), ss, xi(4,1), ss, xi(6,2)} paramindexeslist:={{2,1},{3,1},{4,1},{6,2}} shortformdeltaprimemodadg:={((b(4,4)*b(3,3) - b(4,3)*b(3,4))*xi(2,1))/b(1,1)**3, ss, (b(3,4)*xi(4,1) + b(3,3)*xi(3,1))/b(1,1), ss, (b(4,4)*xi(4,1) + b(4,3)*xi(3,1))/b(1,1), ss, b(1,1)**2*xi(6,2)}$ deltaprimemodg(2,1):=((b(4,4)*b(3,3) - b(4,3)*b(3,4))*xi(2,1))/b(1,1)**3$ deltaprimemodg(3,1):=(b(3,4)*xi(4,1) + b(3,3)*xi(3,1))/b(1,1)$ deltaprimemodg(4,1):=(b(4,4)*xi(4,1) + b(4,3)*xi(3,1))/b(1,1)$ deltaprimemodg(6,2):=b(1,1)**2*xi(6,2)$ det(phi):=(b(4,4)*b(3,3) - b(4,3)*b(3,4))**4/b(1,1)**2$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (b(4,4)*b(3,3) - b(4,3)*b(3,4))*xi(2,1) (-----------------------------------------,0,0,0,0,0), 3 b(1,1) b(3,4)*xi(4,1) + b(3,3)*xi(3,1) (---------------------------------,0,0,0,0,0), b(1,1) b(4,4)*xi(4,1) + b(4,3)*xi(3,1) (---------------------------------,0,0,0,0,0), b(1,1) b(4,4)*xi(4,1) + b(4,3)*xi(3,1) (0,0,---------------------------------, b(1,1) - (b(3,4)*xi(4,1) + b(3,3)*xi(3,1)) --------------------------------------,0,0), b(1,1) 2 (0,b(1,1) *xi(6,2),0,0,0,0)) ************* SUBCASE 1: xi(3,1) and xi(4,1) not both zero. *********$ Then one may suppose xi(3,1):=1 and xi(4,1):=0$ xi(3,1):=1$ xi(4,1):=0$ shortformdeltaprimemodadg:={((b(4,4)*b(3,3) - b(4,3)*b(3,4))*xi(2,1))/b(1,1)**3, ss, b(3,3)/b(1,1), ss, b(4,3)/b(1,1), ss, b(1,1)**2*xi(6,2)}$ deltaprimemodg(2,1):=((b(4,4)*b(3,3) - b(4,3)*b(3,4))*xi(2,1))/b(1,1)**3$ deltaprimemodg(3,1):=b(3,3)/b(1,1)$ deltaprimemodg(4,1):=b(4,3)/b(1,1)$ deltaprimemodg(6,2):=b(1,1)**2*xi(6,2)$ det(phi):=(b(4,4)*b(3,3) - b(4,3)*b(3,4))**4/b(1,1)**2$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0), (b(4,4)*b(3,3) - b(4,3)*b(3,4))*xi(2,1) (-----------------------------------------,0,0,0,0,0), 3 b(1,1) b(3,3) (--------,0,0,0,0,0), b(1,1) b(4,3) (--------,0,0,0,0,0), b(1,1) b(4,3) - b(3,3) (0,0,--------,-----------,0,0), b(1,1) b(1,1) 2 (0,b(1,1) *xi(6,2),0,0,0,0)) To keep xi(3,1):=k,xi(4,1):=0, one has to take:$ b(4,3):=0$ b(3,3):=b(1,1)*k$ shortformdeltaprimemodadg:={(b(4,4)*xi(2,1)*k)/b(1,1)**2, ss, k, ss, 0, ss, b(1,1)**2*xi(6,2)}$ deltaprimemodg(2,1):=(b(4,4)*xi(2,1)*k)/b(1,1)**2$ deltaprimemodg(3,1):=k$ deltaprimemodg(4,1):=0$ deltaprimemodg(6,2):=b(1,1)**2*xi(6,2)$ det(phi):=b(4,4)**4*b(1,1)**2*k**4$ DELTAPRIMEMODADG:= [ 0 0 0 0 0 0] [ ] [ b(4,4)*xi(2,1)*k ] [------------------ 0 0 0 0 0] [ 2 ] [ b(1,1) ] [ ] [ k 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 - k 0 0] [ ] [ 2 ] [ 0 b(1,1) *xi(6,2) 0 0 0 0] Hence, we are reduced in SUBCASE1 to:$ shortformdeltaprime ={epsilon,SS,1,SS,0,SS,eta}$ where epsilon =xi(2,1)=0,1$ where eta =xi(6,2)=0,1$ ************* SUBCASE 2: xi(3,1) = xi(4,1) =0. *********$ xi(3,1):=0$ xi(4,1):=0$ clear b(4,3),b(3,3)$ shortformdeltaprimemodadg:={((b(4,4)*b(3,3) - b(4,3)*b(3,4))*xi(2,1))/b(1,1)**3, ss, 0, ss, 0, ss, b(1,1)**2*xi(6,2)}$ deltaprimemodg(2,1):=((b(4,4)*b(3,3) - b(4,3)*b(3,4))*xi(2,1))/b(1,1)**3$ deltaprimemodg(3,1):=0$ deltaprimemodg(4,1):=0$ deltaprimemodg(6,2):=b(1,1)**2*xi(6,2)$ det(phi):=(b(4,4)*b(3,3) - b(4,3)*b(3,4))**4/b(1,1)**2$ DELTAPRIMEMODADG:= [ 0 0 0 0 0 0] [ ] [ (b(4,4)*b(3,3) - b(4,3)*b(3,4))*xi(2,1) ] [----------------------------------------- 0 0 0 0 0] [ 3 ] [ b(1,1) ] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [ 2 ] [ 0 b(1,1) *xi(6,2) 0 0 0 0] Hence, we are reduced in SUBSUBCASE 2 to:$ shortformdeltaprime ={epsilon,SS,0,SS,0,SS,1}$ where epsilon =xi(2,1)=0,1$ or shortformdeltaprime ={1,SS,0,SS,0,SS,0}$ according to whether or not xi(6,2)=0$