SetDirectory["/www/user/fdahl/papers/Conjugation/"];
<<kappaLib.m
<<Petrov.m

Class XI: (4  11 )

B=(({{0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}}));

V=(({{ lam1, 1, 0, 0, 0, 0}, {0, lam1, 1, 0, 0, 0}, {0, 0, lam1, 1, 0, 0}, {0, 0, 0, lam1, 0, 0}, {0, 0, 0, 0,  lam2, 0}, {0, 0, 0, 0, 0,  lam3}}));

Eigenvalues[V]

W=(({{0, 0, 0, eps1, 0, 0}, {0, 0, eps1, 0, 0, 0}, {0, eps1, 0, 0, 0, 0}, {eps1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, eps2, 0}, {0, 0, 0, 0, 0, eps3}}));

Eigenvalues[W]

eps2 and eps3 have same block size, so we may assume that eps2 <= eps3:

Sort[Eigenvalues[W]/.{eps1->-1,eps2->-1,eps3->-1}]
Sort[Eigenvalues[W]/.{eps1->-1,eps2->-1,eps3->1}]
Sort[Eigenvalues[W]/.{eps1->-1,eps2->1,eps3->1}]
Sort[Eigenvalues[W]/.{eps1->1,eps2->-1,eps3->-1}]
Sort[Eigenvalues[W]/.{eps1->1,eps2->-1,eps3->1}]
Sort[Eigenvalues[W]/.{eps1->1,eps2->1,eps3->1}]

We may assume that eps2 = -1, eps3 = +1

W=W/.{eps2->-1,eps3->1};
Eigenvalues[W]

S=(({{0, 0, 0, 1, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, (1/(Sqrt[2])), (1/(Sqrt[2]))}, {eps1, 0, 0, 0, 0, 0}, {0, eps1, 0, 0, 0, 0}, {0, 0, 0, 0, -(1/(Sqrt[2])), (1/(Sqrt[2]))}}));

Check that S is in set mathcal(S)

Transpose[S].B.S==W

Compute result

res=S.V.Inverse[S];
res//MatrixForm

Petrov[res]

Export notebook as .pdf

NotebookPrint[SelectedNotebook[],"/www/user/fdahl/papers/Conjugation/notebooks/ClassXI.pdf"]