This test verifies when [epsilon,0;0,a;0,0;1,0] is proj equiv to some delta having xi(3,2) neq 0 AND xi(6,3),xi(6,4) still not both zero. 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(3,3), - xi(1,2),0,0), (xi(4,1),xi(4,2), - xi(2,1),xi(3,3) + xi(2,2) - xi(1,1),0,0), (xi(5,1),xi(5,2),xi(5,3),xi(5,4),xi(2,2) + xi(1,1),0), (xi(6,1),xi(6,2),xi(6,3),xi(6,4),xi(4,2) - xi(3,1),xi(3,3) + xi(2,2))) The nonzero adjoints are delta(6,1), delta(6,2), and delta(5,2)+delta(6,4), -delta(5,1)+delta(6,3) Hence -delta(5,2) and delta(6,4) are adjoint related and so are delta(5,1) and delta(6,3) as well. [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 1 0 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 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] [ ] [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] adx4 := [ ] [0 0 0 0 0 0] [ ] [0 0 0 0 0 0] [ ] [-1 0 0 0 0 0] delta is nilpotent if and only if xi(2,2)=-xi(1,1) , xi(3,3)=xi(1,1) and finally the matrix A:=mat((xi(1,1),xi(1,2)),(xi(2,1),xi(2,2))) is nilpotent generic nilpotent derivation : delta:= [xi(1,1) xi(1,2) 0 0 0 0] [ ] [xi(2,1) - xi(1,1) 0 0 0 0] [ ] [xi(3,1) xi(3,2) xi(1,1) - xi(1,2) 0 0] [ ] [xi(4,1) xi(4,2) - xi(2,1) - xi(1,1) 0 0] [ ] [xi(5,1) xi(5,2) xi(5,3) xi(5,4) 0 0] [ ] [xi(6,1) xi(6,2) xi(6,3) xi(6,4) xi(4,2) - xi(3,1) 0] with A:= [xi(1,1) xi(1,2) ] [ ] [xi(2,1) - xi(1,1)] nilpotent The generic automorphism of g_{6,1} as in (art ijac1 section6) : phi:= mat((b(1,1),b(1,2),0,0,0,0), (b(2,1),b(2,2),0,0,0,0), (b(3,1),b(3,2),b(1,1)*u, - b(1,2)*u,0,0), (b(4,1),b(4,2), - b(2,1)*u,b(2,2)*u,0,0), (b(5,1),b(5,2),b(5,3),b(5,4),b(2,2)*b(1,1) - b(2,1)*b(1,2),0), (b(6,1),b(6,2),b(6,3),b(6,4), b(4,2)*b(1,1) - b(4,1)*b(1,2) + b(3,2)*b(2,1) - b(3,1)*b(2,2), u*(b(2,2)*b(1,1) - b(2,1)*b(1,2)))) 4 3 det(phi):=(b(2,2)*b(1,1) - b(2,1)*b(1,2)) *u We consider here in this case 2 the case where A = 0. A:= [0 0] [ ] [0 0] generic nilpotent derivation in the case 1 : delta:= [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [xi(3,1) xi(3,2) 0 0 0 0] [ ] [xi(4,1) xi(4,2) 0 0 0 0] [ ] [xi(5,1) xi(5,2) xi(5,3) xi(5,4) 0 0] [ ] [xi(6,1) xi(6,2) xi(6,3) xi(6,4) xi(4,2) - xi(3,1) 0] by subtracting adjoints one then may suppose xi(5,1)=xi(5,2)=xi(6,1)=xi(6,2)=0 phi:= mat((b(1,1),b(1,2),0,0,0,0), (b(2,1),b(2,2),0,0,0,0), (b(3,1),b(3,2),b(1,1)*u, - b(1,2)*u,0,0), (b(4,1),b(4,2), - b(2,1)*u,b(2,2)*u,0,0), (b(5,1),b(5,2),b(5,3),b(5,4),b(2,2)*b(1,1) - b(2,1)*b(1,2),0), (b(6,1),b(6,2),b(6,3),b(6,4), b(3,2)*b(2,1) - b(3,1)*b(2,2) - b(4,1)*b(1,2) + b(4,2)*b(1,1), (b(2,2)*b(1,1) - b(2,1)*b(1,2))*u)) 4 3 det(phi):=(b(2,2)*b(1,1) - b(2,1)*b(1,2)) *u delta:= [ 0 0 0 0 0 0] [ ] [ 0 0 0 0 0 0] [ ] [xi(3,1) xi(3,2) 0 0 0 0] [ ] [xi(4,1) xi(4,2) 0 0 0 0] [ ] [ 0 0 xi(5,3) xi(5,4) 0 0] [ ] [ 0 0 xi(6,3) xi(6,4) xi(4,2) - xi(3,1) 0] We denote this delta by the shortform shortformdelta:={xi(3,1), xi(3,2), ss, xi(4,1), xi(4,2), ss, xi(5,3), xi(5,4), ss, xi(6,3), xi(6,4)} paramindexeslist:={{3,1},{3,2},{4,1},{4,2},{5,3},{5,4},{6,3},{6,4}} shortformdeltaprimemodadg:={( - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(1,2) - (b(2 ,2)*xi(3,1) - b(2,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), (((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(1,2) - (b(1,2)*xi(3,1) - b(1,1)*xi(3,2))*b (1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ss, (((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(2,2) - (b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b (2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ( - ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(2,2) - (b(1,2)*xi(3,1) - b(1,1)*xi(3,2) )*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ss, (b(2,2)*xi(5,3) + b(2,1)*xi(5,4))/u, (b(1,2)*xi(5,3) + b(1,1)*xi(5,4))/u, ss, ((b(3,2)*b(2,1) - b(3,1)*b(2,2) - b(4,1)*b(1,2) + b(4,2)*b(1,1))*(b(2,2)*xi(5,3) + b(2,1)*xi(5,4)) - ((b(5,4)*b(2,1) + b(5,3)*b(2,2))*(xi(4,2) - xi(3,1)) - (b(2 ,2)*b(1,1) - b(2,1)*b(1,2))*(b(2,2)*xi(6,3) + b(2,1)*xi(6,4)))*u + ((b(3,2)*b(2, 1) - b(3,1)*b(2,2))*b(2,1) - b(4,1)*b(2,2)*b(1,1) + b(4,2)*b(2,1)*b(1,1))*xi(5,4 ) + ((b(3,2)*b(2,1) - b(3,1)*b(2,2) - b(4,1)*b(1,2))*b(2,2) + b(4,2)*b(2,1)*b(1, 2))*xi(5,3) + ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(5,4) + (b(2,2)*xi(3,1) - b(2, 1)*xi(3,2))*b(5,3))*u)/((b(2,2)*b(1,1) - b(2,1)*b(1,2))*u), (((b(3,2)*b(1,1) - b(3,1)*b(1,2))*b(2,1) - b(4,1)*b(1,2)*b(1,1) + b(4,2)*b(1,1) **2)*xi(5,4) + ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(5,4) + (b(1,2)*xi(3,1) - b(1 ,1)*xi(3,2))*b(5,3))*u + ((b(3,2)*b(1,1) - b(3,1)*b(1,2))*b(2,2) - b(4,1)*b(1,2) **2 + b(4,2)*b(1,2)*b(1,1))*xi(5,3) + (b(3,2)*b(2,1) - b(3,1)*b(2,2) - b(4,1)*b( 1,2) + b(4,2)*b(1,1))*(b(1,2)*xi(5,3) + b(1,1)*xi(5,4)) - ((b(5,4)*b(1,1) + b(5, 3)*b(1,2))*(xi(4,2) - xi(3,1)) - (b(2,2)*b(1,1) - b(2,1)*b(1,2))*(b(1,2)*xi(6,3) + b(1,1)*xi(6,4)))*u)/((b(2,2)*b(1,1) - b(2,1)*b(1,2))*u)}$ deltaprimemodg(3,1):=( - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(1,2) - (b(2,2)*xi( 3,1) - b(2,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(3,2):=(((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(1,2) - (b(1,2)*xi(3,1 ) - b(1,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(4,1):=(((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(2,2) - (b(2,2)*xi(3,1 ) - b(2,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(4,2):=( - ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(2,2) - (b(1,2)*xi( 3,1) - b(1,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(5,3):=(b(2,2)*xi(5,3) + b(2,1)*xi(5,4))/u$ deltaprimemodg(5,4):=(b(1,2)*xi(5,3) + b(1,1)*xi(5,4))/u$ deltaprimemodg(6,3):=((b(3,2)*b(2,1) - b(3,1)*b(2,2) - b(4,1)*b(1,2) + b(4,2)*b( 1,1))*(b(2,2)*xi(5,3) + b(2,1)*xi(5,4)) - ((b(5,4)*b(2,1) + b(5,3)*b(2,2))*(xi(4 ,2) - xi(3,1)) - (b(2,2)*b(1,1) - b(2,1)*b(1,2))*(b(2,2)*xi(6,3) + b(2,1)*xi(6,4 )))*u + ((b(3,2)*b(2,1) - b(3,1)*b(2,2))*b(2,1) - b(4,1)*b(2,2)*b(1,1) + b(4,2)* b(2,1)*b(1,1))*xi(5,4) + ((b(3,2)*b(2,1) - b(3,1)*b(2,2) - b(4,1)*b(1,2))*b(2,2) + b(4,2)*b(2,1)*b(1,2))*xi(5,3) + ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(5,4) + ( b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b(5,3))*u)/((b(2,2)*b(1,1) - b(2,1)*b(1,2))*u)$ deltaprimemodg(6,4):=(((b(3,2)*b(1,1) - b(3,1)*b(1,2))*b(2,1) - b(4,1)*b(1,2)*b( 1,1) + b(4,2)*b(1,1)**2)*xi(5,4) + ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(5,4) + ( b(1,2)*xi(3,1) - b(1,1)*xi(3,2))*b(5,3))*u + ((b(3,2)*b(1,1) - b(3,1)*b(1,2))*b( 2,2) - b(4,1)*b(1,2)**2 + b(4,2)*b(1,2)*b(1,1))*xi(5,3) + (b(3,2)*b(2,1) - b(3,1 )*b(2,2) - b(4,1)*b(1,2) + b(4,2)*b(1,1))*(b(1,2)*xi(5,3) + b(1,1)*xi(5,4)) - (( b(5,4)*b(1,1) + b(5,3)*b(1,2))*(xi(4,2) - xi(3,1)) - (b(2,2)*b(1,1) - b(2,1)*b(1 ,2))*(b(1,2)*xi(6,3) + b(1,1)*xi(6,4)))*u)/((b(2,2)*b(1,1) - b(2,1)*b(1,2))*u)$ det(phi):=(b(2,2)*b(1,1) - b(2,1)*b(1,2))**4*u**3$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(( - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(1,2) - (b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),(( (b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(1,2) - (b(1,2)*xi(3,1) - b(1,1)*xi(3,2))*b(1 ,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),0,0,0,0),((((b(2,2)*xi(4,1) - b(2,1)*xi( 4,2))*b(2,2) - (b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2 ,1)*b(1,2)),( - ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(2,2) - (b(1,2)*xi(3,1) - b( 1,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),0,0,0,0),(0,0,(b(2,2)* xi(5,3) + b(2,1)*xi(5,4))/u,(b(1,2)*xi(5,3) + b(1,1)*xi(5,4))/u,0,0),(0,0,((b(3, 2)*b(2,1) - b(3,1)*b(2,2) - b(4,1)*b(1,2) + b(4,2)*b(1,1))*(b(2,2)*xi(5,3) + b(2 ,1)*xi(5,4)) - ((b(5,4)*b(2,1) + b(5,3)*b(2,2))*(xi(4,2) - xi(3,1)) - (b(2,2)*b( 1,1) - b(2,1)*b(1,2))*(b(2,2)*xi(6,3) + b(2,1)*xi(6,4)))*u + ((b(3,2)*b(2,1) - b (3,1)*b(2,2))*b(2,1) - b(4,1)*b(2,2)*b(1,1) + b(4,2)*b(2,1)*b(1,1))*xi(5,4) + (( b(3,2)*b(2,1) - b(3,1)*b(2,2) - b(4,1)*b(1,2))*b(2,2) + b(4,2)*b(2,1)*b(1,2))*xi (5,3) + ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(5,4) + (b(2,2)*xi(3,1) - b(2,1)*xi( 3,2))*b(5,3))*u)/((b(2,2)*b(1,1) - b(2,1)*b(1,2))*u),(((b(3,2)*b(1,1) - b(3,1)*b (1,2))*b(2,1) - b(4,1)*b(1,2)*b(1,1) + b(4,2)*b(1,1)**2)*xi(5,4) + ((b(1,2)*xi(4 ,1) - b(1,1)*xi(4,2))*b(5,4) + (b(1,2)*xi(3,1) - b(1,1)*xi(3,2))*b(5,3))*u + ((b (3,2)*b(1,1) - b(3,1)*b(1,2))*b(2,2) - b(4,1)*b(1,2)**2 + b(4,2)*b(1,2)*b(1,1))* xi(5,3) + (b(3,2)*b(2,1) - b(3,1)*b(2,2) - b(4,1)*b(1,2) + b(4,2)*b(1,1))*(b(1,2 )*xi(5,3) + b(1,1)*xi(5,4)) - ((b(5,4)*b(1,1) + b(5,3)*b(1,2))*(xi(4,2) - xi(3,1 )) - (b(2,2)*b(1,1) - b(2,1)*b(1,2))*(b(1,2)*xi(6,3) + b(1,1)*xi(6,4)))*u)/((b(2 ,2)*b(1,1) - b(2,1)*b(1,2))*u),(xi(4,2) - xi(3,1))*u,0))$ ********* Suppose that xi(5,3) and xi(5,4) are both zero.$ xi(5,3):=0$ xi(5,4):=0$ shortformdeltaprimemodadg:={( - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(1,2) - (b(2 ,2)*xi(3,1) - b(2,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), (((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(1,2) - (b(1,2)*xi(3,1) - b(1,1)*xi(3,2))*b (1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ss, (((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(2,2) - (b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b (2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ( - ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(2,2) - (b(1,2)*xi(3,1) - b(1,1)*xi(3,2) )*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ss, 0, 0, ss, ( - ((b(5,4)*b(2,1) + b(5,3)*b(2,2))*(xi(4,2) - xi(3,1)) - (b(2,2)*b(1,1) - b(2, 1)*b(1,2))*(b(2,2)*xi(6,3) + b(2,1)*xi(6,4)) - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2) )*b(5,4) + (b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b(5,3))))/(b(2,2)*b(1,1) - b(2,1)*b (1,2)), ( - ((b(5,4)*b(1,1) + b(5,3)*b(1,2))*(xi(4,2) - xi(3,1)) - (b(2,2)*b(1,1) - b(2, 1)*b(1,2))*(b(1,2)*xi(6,3) + b(1,1)*xi(6,4)) - ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2) )*b(5,4) + (b(1,2)*xi(3,1) - b(1,1)*xi(3,2))*b(5,3))))/(b(2,2)*b(1,1) - b(2,1)*b (1,2))}$ deltaprimemodg(3,1):=( - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(1,2) - (b(2,2)*xi( 3,1) - b(2,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(3,2):=(((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(1,2) - (b(1,2)*xi(3,1 ) - b(1,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(4,1):=(((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(2,2) - (b(2,2)*xi(3,1 ) - b(2,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(4,2):=( - ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(2,2) - (b(1,2)*xi( 3,1) - b(1,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(5,3):=0$ deltaprimemodg(5,4):=0$ deltaprimemodg(6,3):=( - ((b(5,4)*b(2,1) + b(5,3)*b(2,2))*(xi(4,2) - xi(3,1)) - (b(2,2)*b(1,1) - b(2,1)*b(1,2))*(b(2,2)*xi(6,3) + b(2,1)*xi(6,4)) - ((b(2,2)*xi( 4,1) - b(2,1)*xi(4,2))*b(5,4) + (b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b(5,3))))/(b(2 ,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(6,4):=( - ((b(5,4)*b(1,1) + b(5,3)*b(1,2))*(xi(4,2) - xi(3,1)) - (b(2,2)*b(1,1) - b(2,1)*b(1,2))*(b(1,2)*xi(6,3) + b(1,1)*xi(6,4)) - ((b(1,2)*xi( 4,1) - b(1,1)*xi(4,2))*b(5,4) + (b(1,2)*xi(3,1) - b(1,1)*xi(3,2))*b(5,3))))/(b(2 ,2)*b(1,1) - b(2,1)*b(1,2))$ det(phi):=(b(2,2)*b(1,1) - b(2,1)*b(1,2))**4*u**3$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(( - ((b(2,2)*xi(4,1) - b(2,1)*xi(4,2))*b(1,2) - (b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b(1,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),(( (b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(1,2) - (b(1,2)*xi(3,1) - b(1,1)*xi(3,2))*b(1 ,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),0,0,0,0),((((b(2,2)*xi(4,1) - b(2,1)*xi( 4,2))*b(2,2) - (b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2 ,1)*b(1,2)),( - ((b(1,2)*xi(4,1) - b(1,1)*xi(4,2))*b(2,2) - (b(1,2)*xi(3,1) - b( 1,1)*xi(3,2))*b(2,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),0,0,0,0),(0,0,0,0,0,0), (0,0,( - ((b(5,4)*b(2,1) + b(5,3)*b(2,2))*(xi(4,2) - xi(3,1)) - (b(2,2)*b(1,1) - b(2,1)*b(1,2))*(b(2,2)*xi(6,3) + b(2,1)*xi(6,4)) - ((b(2,2)*xi(4,1) - b(2,1)*xi (4,2))*b(5,4) + (b(2,2)*xi(3,1) - b(2,1)*xi(3,2))*b(5,3))))/(b(2,2)*b(1,1) - b(2 ,1)*b(1,2)),( - ((b(5,4)*b(1,1) + b(5,3)*b(1,2))*(xi(4,2) - xi(3,1)) - (b(2,2)*b (1,1) - b(2,1)*b(1,2))*(b(1,2)*xi(6,3) + b(1,1)*xi(6,4)) - ((b(1,2)*xi(4,1) - b( 1,1)*xi(4,2))*b(5,4) + (b(1,2)*xi(3,1) - b(1,1)*xi(3,2))*b(5,3))))/(b(2,2)*b(1,1 ) - b(2,1)*b(1,2)),(xi(4,2) - xi(3,1))*u,0))$ ******Suppose that xi(6,3) and xi(6,4) are noth both zero:$ xi(6,3):=1$ xi(6,4):=0$ shortformdelta:={xi(3,1), 0, ss, 0, xi(4,2), ss, 0, 0, ss, 1, 0}$ Recall that xi(3,1)=epsilon=0,1 and xi(4,2)=a NEQ -xi(3,1)$ shortformdeltaprimemodadg:={((b(2,2)*b(1,1)*xi(3,1) + b(2,1)*b(1,2)*xi(4,2))*u)/ (b(2,2)*b(1,1) - b(2,1)*b(1,2)), ( - (xi(4,2) + xi(3,1))*b(1,2)*b(1,1)*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ss, ( - (xi(4,2) + xi(3,1))*b(2,2)*b(2,1)*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ((b(2,2)*b(1,1)*xi(4,2) + b(2,1)*b(1,2)*xi(3,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1, 2)), ss, 0, 0, ss, ( - (2*b(5,4)*b(2,1)*xi(4,2) - b(5,4)*b(2,1)*xi(3,1) + b(5,3)*b(2,2)*xi(4,2) - 2 *b(5,3)*b(2,2)*xi(3,1) - b(2,2)**2*b(1,1) + b(2,2)*b(2,1)*b(1,2)))/(b(2,2)*b(1,1 ) - b(2,1)*b(1,2)), ( - (2*b(5,4)*b(1,1)*xi(4,2) - b(5,4)*b(1,1)*xi(3,1) + b(5,3)*b(1,2)*xi(4,2) - 2 *b(5,3)*b(1,2)*xi(3,1) - b(2,2)*b(1,2)*b(1,1) + b(2,1)*b(1,2)**2))/(b(2,2)*b(1,1 ) - b(2,1)*b(1,2))}$ deltaprimemodg(3,1):=((b(2,2)*b(1,1)*xi(3,1) + b(2,1)*b(1,2)*xi(4,2))*u)/(b(2,2) *b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(3,2):=( - (xi(4,2) + xi(3,1))*b(1,2)*b(1,1)*u)/(b(2,2)*b(1,1) - b (2,1)*b(1,2))$ deltaprimemodg(4,1):=( - (xi(4,2) + xi(3,1))*b(2,2)*b(2,1)*u)/(b(2,2)*b(1,1) - b (2,1)*b(1,2))$ deltaprimemodg(4,2):=((b(2,2)*b(1,1)*xi(4,2) + b(2,1)*b(1,2)*xi(3,1))*u)/(b(2,2) *b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(5,3):=0$ deltaprimemodg(5,4):=0$ deltaprimemodg(6,3):=( - (2*b(5,4)*b(2,1)*xi(4,2) - b(5,4)*b(2,1)*xi(3,1) + b(5, 3)*b(2,2)*xi(4,2) - 2*b(5,3)*b(2,2)*xi(3,1) - b(2,2)**2*b(1,1) + b(2,2)*b(2,1)*b (1,2)))/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(6,4):=( - (2*b(5,4)*b(1,1)*xi(4,2) - b(5,4)*b(1,1)*xi(3,1) + b(5, 3)*b(1,2)*xi(4,2) - 2*b(5,3)*b(1,2)*xi(3,1) - b(2,2)*b(1,2)*b(1,1) + b(2,1)*b(1, 2)**2))/(b(2,2)*b(1,1) - b(2,1)*b(1,2))$ det(phi):=(b(2,2)*b(1,1) - b(2,1)*b(1,2))**4*u**3$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(((b(2,2)*b(1,1)*xi(3,1) + b(2,1)*b(1,2)*xi(4,2) )*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),( - (xi(4,2) + xi(3,1))*b(1,2)*b(1,1)*u)/(b (2,2)*b(1,1) - b(2,1)*b(1,2)),0,0,0,0),(( - (xi(4,2) + xi(3,1))*b(2,2)*b(2,1)*u) /(b(2,2)*b(1,1) - b(2,1)*b(1,2)),((b(2,2)*b(1,1)*xi(4,2) + b(2,1)*b(1,2)*xi(3,1) )*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),0,0,0,0),(0,0,0,0,0,0),(0,0,( - (2*b(5,4)*b (2,1)*xi(4,2) - b(5,4)*b(2,1)*xi(3,1) + b(5,3)*b(2,2)*xi(4,2) - 2*b(5,3)*b(2,2)* xi(3,1) - b(2,2)**2*b(1,1) + b(2,2)*b(2,1)*b(1,2)))/(b(2,2)*b(1,1) - b(2,1)*b(1, 2)),( - (2*b(5,4)*b(1,1)*xi(4,2) - b(5,4)*b(1,1)*xi(3,1) + b(5,3)*b(1,2)*xi(4,2) - 2*b(5,3)*b(1,2)*xi(3,1) - b(2,2)*b(1,2)*b(1,1) + b(2,1)*b(1,2)**2))/(b(2,2)*b (1,1) - b(2,1)*b(1,2)),(xi(4,2) - xi(3,1))*u,0))$ Hence as xi(4,2)+xi(3,1) neq 0, we are back to the case xi(3,2) neq 0.$ However, it is not automatic that we stay with xi(6,3), xi(6,4) not both zero$ since in reducparautommodg6_1case2N4.red we externally decreed b(5,3)=b(5,4)=0$ SUBCASE 1: SUPPOSE: 2*xi(4,2) NEQ xi(3,1).$ that is a NEQ epsilon/2.$ Then, with b(1,1) NEQ 0, to get deltaprimemodg(6,4)=0, one has to take:$ b(5,4):=((b(2,2)*b(1,1) - b(2,1)*b(1,2) - (xi(4,2) - 2*xi(3,1))*b(5,3))*b(1,2))/ ((2*xi(4,2) - xi(3,1))*b(1,1))$ shortformdeltaprimemodadg:={((b(2,2)*b(1,1)*xi(3,1) + b(2,1)*b(1,2)*xi(4,2))*u)/ (b(2,2)*b(1,1) - b(2,1)*b(1,2)), ( - (xi(4,2) + xi(3,1))*b(1,2)*b(1,1)*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ss, ( - (xi(4,2) + xi(3,1))*b(2,2)*b(2,1)*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ((b(2,2)*b(1,1)*xi(4,2) + b(2,1)*b(1,2)*xi(3,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1, 2)), ss, 0, 0, ss, ( - (b(5,3)*xi(4,2) - 2*b(5,3)*xi(3,1) - b(2,2)*b(1,1) + b(2,1)*b(1,2)))/b(1,1), 0}$ deltaprimemodg(3,1):=((b(2,2)*b(1,1)*xi(3,1) + b(2,1)*b(1,2)*xi(4,2))*u)/(b(2,2) *b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(3,2):=( - (xi(4,2) + xi(3,1))*b(1,2)*b(1,1)*u)/(b(2,2)*b(1,1) - b (2,1)*b(1,2))$ deltaprimemodg(4,1):=( - (xi(4,2) + xi(3,1))*b(2,2)*b(2,1)*u)/(b(2,2)*b(1,1) - b (2,1)*b(1,2))$ deltaprimemodg(4,2):=((b(2,2)*b(1,1)*xi(4,2) + b(2,1)*b(1,2)*xi(3,1))*u)/(b(2,2) *b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(5,3):=0$ deltaprimemodg(5,4):=0$ deltaprimemodg(6,3):=( - (b(5,3)*xi(4,2) - 2*b(5,3)*xi(3,1) - b(2,2)*b(1,1) + b( 2,1)*b(1,2)))/b(1,1)$ deltaprimemodg(6,4):=0$ det(phi):=(b(2,2)*b(1,1) - b(2,1)*b(1,2))**4*u**3$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(((b(2,2)*b(1,1)*xi(3,1) + b(2,1)*b(1,2)*xi(4,2) )*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),( - (xi(4,2) + xi(3,1))*b(1,2)*b(1,1)*u)/(b (2,2)*b(1,1) - b(2,1)*b(1,2)),0,0,0,0),(( - (xi(4,2) + xi(3,1))*b(2,2)*b(2,1)*u) /(b(2,2)*b(1,1) - b(2,1)*b(1,2)),((b(2,2)*b(1,1)*xi(4,2) + b(2,1)*b(1,2)*xi(3,1) )*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),0,0,0,0),(0,0,0,0,0,0),(0,0,( - (b(5,3)*xi( 4,2) - 2*b(5,3)*xi(3,1) - b(2,2)*b(1,1) + b(2,1)*b(1,2)))/b(1,1),0,(xi(4,2) - xi (3,1))*u,0))$ SUBCASE 1.1: SUPPOSE MOREOVER : xi(4,2) NEQ 2*xi(3,1).$ that is : a NEQ 2*epsilon.$ Then, with b(1,1) NEQ 0, to get deltaprimemodg(6,3)=1, one has to take:$ b(5,4):=b(1,2)/(2*xi(4,2) - xi(3,1))$ shortformdeltaprimemodadg:={((b(2,2)*b(1,1)*xi(3,1) + b(2,1)*b(1,2)*xi(4,2))*u)/ (b(2,2)*b(1,1) - b(2,1)*b(1,2)), ( - (xi(4,2) + xi(3,1))*b(1,2)*b(1,1)*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ss, ( - (xi(4,2) + xi(3,1))*b(2,2)*b(2,1)*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)), ((b(2,2)*b(1,1)*xi(4,2) + b(2,1)*b(1,2)*xi(3,1))*u)/(b(2,2)*b(1,1) - b(2,1)*b(1, 2)), ss, 0, 0, ss, 1, 0}$ deltaprimemodg(3,1):=((b(2,2)*b(1,1)*xi(3,1) + b(2,1)*b(1,2)*xi(4,2))*u)/(b(2,2) *b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(3,2):=( - (xi(4,2) + xi(3,1))*b(1,2)*b(1,1)*u)/(b(2,2)*b(1,1) - b (2,1)*b(1,2))$ deltaprimemodg(4,1):=( - (xi(4,2) + xi(3,1))*b(2,2)*b(2,1)*u)/(b(2,2)*b(1,1) - b (2,1)*b(1,2))$ deltaprimemodg(4,2):=((b(2,2)*b(1,1)*xi(4,2) + b(2,1)*b(1,2)*xi(3,1))*u)/(b(2,2) *b(1,1) - b(2,1)*b(1,2))$ deltaprimemodg(5,3):=0$ deltaprimemodg(5,4):=0$ deltaprimemodg(6,3):=1$ deltaprimemodg(6,4):=0$ det(phi):=(b(2,2)*b(1,1) - b(2,1)*b(1,2))**4*u**3$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(((b(2,2)*b(1,1)*xi(3,1) + b(2,1)*b(1,2)*xi(4,2) )*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),( - (xi(4,2) + xi(3,1))*b(1,2)*b(1,1)*u)/(b (2,2)*b(1,1) - b(2,1)*b(1,2)),0,0,0,0),(( - (xi(4,2) + xi(3,1))*b(2,2)*b(2,1)*u) /(b(2,2)*b(1,1) - b(2,1)*b(1,2)),((b(2,2)*b(1,1)*xi(4,2) + b(2,1)*b(1,2)*xi(3,1) )*u)/(b(2,2)*b(1,1) - b(2,1)*b(1,2)),0,0,0,0),(0,0,0,0,0,0),(0,0,1,0,(xi(4,2) - xi(3,1))*u,0))$ ********** Hence as a +epsilon = xi(4,2)+xi(3,1) neq 0,$ In subcase 1.1. $ when 2*a NEQ epsilon AND a NEQ 2*epsilon$ we are back to the case xi(3,2) neq 0$ and we stay with xi(6,3)= 1 , xi(6,4) =0.$ ********** Now : SUBCASE 1.2 :$ that is : a = 2*epsilon.$ Then necessarily 2*epsilon NEQ epsilon implies epsilon=1, a= 1$ It is the case considered in test1case2N52$ ********** Finally : CASE 2 :$ a = epsilon/2.$ As a NEQ - epsilon, we have necessarily epsilon = 1, a =1/2.$ This is the case2IX8, which we already now to be nonisomorphic to $ some delta having xi(3,2) neq 0 AND xi(6,3)=1, xi(6,4)=0.$ Let us check that.$ clear b(5,4),b(5,3)$ shortformdeltaprimemodadg:={((2*b(2,2)*b(1,1) + b(2,1)*b(1,2))*u)/(2*(b(2,2)*b(1 ,1) - b(2,1)*b(1,2))), ( - 3*b(1,2)*b(1,1)*u)/(2*(b(2,2)*b(1,1) - b(2,1)*b(1,2))), ss, ( - 3*b(2,2)*b(2,1)*u)/(2*(b(2,2)*b(1,1) - b(2,1)*b(1,2))), ((b(2,2)*b(1,1) + 2*b(2,1)*b(1,2))*u)/(2*(b(2,2)*b(1,1) - b(2,1)*b(1,2))), ss, 0, 0, ss, ((3*b(5,3) + 2*b(2,2)*b(1,1) - 2*b(2,1)*b(1,2))*b(2,2))/(2*(b(2,2)*b(1,1) - b(2, 1)*b(1,2))), ((3*b(5,3) + 2*b(2,2)*b(1,1) - 2*b(2,1)*b(1,2))*b(1,2))/(2*(b(2,2)*b(1,1) - b(2, 1)*b(1,2)))}$ deltaprimemodg(3,1):=((2*b(2,2)*b(1,1) + b(2,1)*b(1,2))*u)/(2*(b(2,2)*b(1,1) - b (2,1)*b(1,2)))$ deltaprimemodg(3,2):=( - 3*b(1,2)*b(1,1)*u)/(2*(b(2,2)*b(1,1) - b(2,1)*b(1,2)))$ deltaprimemodg(4,1):=( - 3*b(2,2)*b(2,1)*u)/(2*(b(2,2)*b(1,1) - b(2,1)*b(1,2)))$ deltaprimemodg(4,2):=((b(2,2)*b(1,1) + 2*b(2,1)*b(1,2))*u)/(2*(b(2,2)*b(1,1) - b (2,1)*b(1,2)))$ deltaprimemodg(5,3):=0$ deltaprimemodg(5,4):=0$ deltaprimemodg(6,3):=((3*b(5,3) + 2*b(2,2)*b(1,1) - 2*b(2,1)*b(1,2))*b(2,2))/(2* (b(2,2)*b(1,1) - b(2,1)*b(1,2)))$ deltaprimemodg(6,4):=((3*b(5,3) + 2*b(2,2)*b(1,1) - 2*b(2,1)*b(1,2))*b(1,2))/(2* (b(2,2)*b(1,1) - b(2,1)*b(1,2)))$ det(phi):=(b(2,2)*b(1,1) - b(2,1)*b(1,2))**4*u**3$ DELTAPRIMEMODADG:= mat((0,0,0,0,0,0),(0,0,0,0,0,0),(((2*b(2,2)*b(1,1) + b(2,1)*b(1,2))*u)/(2*(b(2,2 )*b(1,1) - b(2,1)*b(1,2))),( - 3*b(1,2)*b(1,1)*u)/(2*(b(2,2)*b(1,1) - b(2,1)*b(1 ,2))),0,0,0,0),(( - 3*b(2,2)*b(2,1)*u)/(2*(b(2,2)*b(1,1) - b(2,1)*b(1,2))),((b(2 ,2)*b(1,1) + 2*b(2,1)*b(1,2))*u)/(2*(b(2,2)*b(1,1) - b(2,1)*b(1,2))),0,0,0,0),(0 ,0,0,0,0,0),(0,0,((3*b(5,3) + 2*b(2,2)*b(1,1) - 2*b(2,1)*b(1,2))*b(2,2))/(2*(b(2 ,2)*b(1,1) - b(2,1)*b(1,2))),((3*b(5,3) + 2*b(2,2)*b(1,1) - 2*b(2,1)*b(1,2))*b(1 ,2))/(2*(b(2,2)*b(1,1) - b(2,1)*b(1,2))),( - u)/2,0))$ we see that deltaprime(6,3)=deltaprime(6,4), hence it is not possible$ to get the first = 1 and the second equal = 0$