SetDirectory["~/KappaLib"];
<<kappaLib-1.2.m

Metaclass I representation of Lorentz Hodge operator + Axion

kappa=emMatrixToKappa[(({{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}}))];
kappa=kappa/.{a2->a1,a3->a1,b3->b1,b2->b1};

emKappaToMatrix[kappa]//MatrixForm

g=DiagonalMatrix[{-1,1,1,1}];
kappaAlt=-b1 emHodge[g] + a1 emIdentityKappa[];

Union[Flatten[kappaAlt-kappa]]