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

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=(({{ LL1, 1, 0, 0, 0, 0}, {0, LL1, 1, 0, 0, 0}, {0, 0, LL1, 0, 0, 0}, {0, 0, 0, LL2, 1, 0}, {0, 0, 0, 0,  LL2, 1}, {0, 0, 0, 0, 0,  LL2}}));
W=(({{0, 0, -1, 0, 0, 0}, {0, -1, 0, 0, 0, 0}, {-1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 1, 0, 0}}));

Solve S manually

mat=(({{lam1, 0, 0, 0, 0, 0}, {(1/(Sqrt[2])), ((lam2+lam1)/2), -(1/(Sqrt[2])), 0, (1/2) (-lam2+lam1), 0}, {0, 0, lam2, 0, 0, 0}, {0, (1/(Sqrt[2])), 0, lam1, (1/(Sqrt[2])), 0}, {(1/(Sqrt[2])), (1/2) (-lam2+lam1), (1/(Sqrt[2])), 0, ((lam1+lam2)/2), 0}, {0, (1/(Sqrt[2])), 0, 0, -(1/(Sqrt[2])), lam2}}));
Petrov[mat]
S=Table[ToExpression["s"<>ToString[i]<>ToString[j]],{i,1,6},{j,1,6}];
j1=Flatten[Transpose[S].B.S-W];
j2=Flatten[mat.S-S.V];

exp=Join[j1,j2];

sub={lam1->LL2,lam2->LL1,s14->0,s15->0,s31->0,s32->0,s34->0,s11->0,s35->0,s12->0,s21->s51,s24->-s54,s51->0,s54->0,s36->0,s64->0,s13->0,s41->0,s22->-s52,s42->0,s25->s55,s61->-Sqrt[2]s52,s65->0,s66->0,s43->0,s26->s56,s44->Sqrt[2]s55,s55->s16/Sqrt[2],s23->-s53,s52->s33/Sqrt[2],s62->-Sqrt[2]s53,s53->0,s45->Sqrt[2]s56,s63->0,s56->0,s33->1,s16->1,s46->0};
Simplify[Sort[Union[exp//.sub],emSize[#1]≤ emSize[#2]&]]//MatrixForm
Simplify[S//.sub] //MatrixForm

(* export .pdf *)
NotebookPrint[SelectedNotebook[],"/www/user/fdahl/papers/Conjugation/notebooks/ClassXX_Solve.pdf"]