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

Class X: (4  1bar(1) )

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,  sigma1, tau1}, {0, 0, 0, 0, -tau1,  sigma1}}));

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, 0, 1}, {0, 0, 0, 0, 1, 0}}));

Value of eps1 can be +1/-1.

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

S=(({{0, 0, 0, 1, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 1}, {eps1, 0, 0, 0, 0, 0}, {0, eps1, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0}}));

Check that S is in set mathcal(S)

Transpose[S].B.S==W

Compute result

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

Petrov[res]

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