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

Class IV: (11  1bar(1) 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=(({{ lambda1, 0, 0, 0, 0, 0}, {0, lambda2, 0, 0, 0, 0}, {0, 0, sigma1, tau1, 0, 0}, {0, 0, -tau1, sigma1, 0, 0}, {0, 0, 0, 0,  sigma2, tau2}, {0, 0, 0, 0, -tau2,  sigma2}}));

Eigenvalues[V]

W=(({{eps1, 0, 0, 0, 0, 0}, {0, eps2, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 1, 0}}));

Eigenvalues[W]

Since eps1 and eps2 have same block size, we assume that eps1 <= eps2. 
We also know that W should have signature (+++---).

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

Only sign possibility is eps1 = -1, eps2 = +1

W=W/.{eps1->-1,eps2->1};
W//MatrixForm
Eigenvalues[W]

(* Permutation 5,5 with leading B *)
S=(({{0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 1}, {(1/(Sqrt[2])), (1/(Sqrt[2])), 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 1, 0}, {-(1/(Sqrt[2])), (1/(Sqrt[2])), 0, 0, 0, 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]

Export notebook as .pdf

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