Coordinate conjugations for Metaclass I

SetDirectory["~/writing/WIP/Conjugation/"]
<<kappaLib.m

mat1=(({{a1, 0, 0, -b1, 0, 0}, {0, a2, 0, 0, -b2, 0}, {0, 0, a3, 0, 0, -b3}, {b1, 0, 0, a1, 0, 0}, {0, b2, 0, 0, a2, 0}, {0, 0, b3, 0, 0, a3}}));

H2=(({{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}}));
H3=(({{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}}));
kappa1=emMatrixToKappa[mat1];
kappa2=emMatrixToKappa[H2.mat1.H2];
kappa3=emMatrixToKappa[H3.mat1.H3];

Union[Flatten[kappa2-kappa3]]

L=DiagonalMatrix[{-1,1,1,1}];
kappa2a =emCoordinateChange[kappa2,L];

Union[Flatten[kappa2a-kappa1]]