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

Define Metaclass I

kappa=emMatrixToKappa[(({{a1, 0, 0, -b1, 0, 0}, {0, a2, 0, 0, -b2, 0}, {0, 0, a3, 0, 0, a4}, {b1, 0, 0, a1, 0, 0}, {0, b2, 0, 0, a2, 0}, {0, 0, a4, 0, 0, a3}}))];
sub={a2->a1,b2->b1};
kappa=kappa//.sub;

D1 =( (a1-a3)^2+b1^2-a4^2 )/(b1 a4)

Case: a1 = a3

Define bivector A and metric g

Abivector=(({{0, 0, 0, 1}, {0, 0, 0, 0}, {0, 0, 0, 0}, {-1, 0, 0, 0}}));
Metric = Inverse[(({{1, 0, 0, 0}, {0, Psi, 0, 0}, {0, 0, Psi, 0}, {0, 0, 0, -1}}))];

Check that metric has Lorentz signature

Det[Metric]

Formulate equations that should be satisfied

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

sub={
Psi->a4/b1,
a3->a1,
C1->((b1/Psi)/SqrtAbsDetG),
C2->a1,
rho->(((a4^2)+(b1^2))/(2 a4))
};
show[simp[eqs//.sub]]

Case: a1 != a3

Define A and metric g

Abivector=(({{0, 0, 0, ((-a1+a3)/(2 rho SqrtAbsXi))}, {0, 0, SqrtAbsXi, 0}, {0, -SqrtAbsXi, 0, 0}, {-((-a1+a3)/(2 rho SqrtAbsXi)), 0, 0, 0}}));
Metric=Inverse[(({{1, 0, 0, 0}, {0, Psi, 0, 0}, {0, 0, Psi, 0}, {0, 0, 0, -1}}))];

Show that the equations are satisfied when suitable choice of parameters:

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

sub={
C2->a1,
C1->b1 ((1/Psi)/SqrtAbsDetG)
};
show[simp[eqs//.sub]]

By definition of rho and Xi, these equations are satisfied.