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

Define Metaclass II

kappa=emMatrixToKappa[(({{a1, -b1, 0, 0, 0, 0}, {b1, a1, 0, 0, 0, 0}, {0, 0, a2, 0, 0, -b2}, {0, 1, 0, a1, b1, 0}, {1, 0, 0, -b1, a1, 0}, {0, 0, b2, 0, 0, a2}}))];

sub={a2->a1,b2->b1};
kappa=kappa//.sub;;

emKappaToMatrix[kappa]//MatrixForm

Define A and metric g

Abivector=(({{0, 1,  1, 0}, {-1, 0, 0, 0}, {-1, 0, 0, 0}, {0, 0, 0, 0}}));
Metric= Inverse[(({{-1, 0, 0, b1}, {0, -b1, 0, 0}, {0, 0, -b1, 0}, {b1, 0, 0, 0}}))];
C1=-((1/b1)/SqrtAbsDetG);
C2=a1 ;
rho=1/2 ;

Formulate equations that should be satisfied

kappaAlt=C1 emQMedium[SqrtAbsDetG,Inverse[Metric]] + emBiProduct[rho,Abivector, Abivector] +C2 emIdentityKappa[];
eqs=Union[Flatten[FullSimplify[kappa-kappaAlt]]]

Check that metric has a Lorentz signature

Det[Metric]

Extra: Alternative expression for metric and A

Define A and metric g

Abivector=(({{0, 1, - 1, 0}, {-1, 0, 0, 0}, { 1, 0, 0, 0}, {0, 0, 0, 0}}));
Metric= Inverse[(({{1, 0, 0, b1}, {0, -b1, 0, 0}, {0, 0, -b1, 0}, {b1, 0, 0, 0}}))];
C1=-((1/b1)/SqrtAbsDetG);
C2=a1 ;
rho=-1/2;

Formulate equations that should be satisfied

kappaAlt=C1 emQMedium[SqrtAbsDetG,Inverse[Metric]] + emBiProduct[rho,Abivector, Abivector] +C2 emIdentityKappa[];
eqs=Union[Flatten[FullSimplify[kappa-kappaAlt]]]

Check that metric has a Lorentz signature

Det[Metric]