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

Define Metaclass V with parameters:

     alpha_i in R,    beta_i in R\0,    and    beta_i all have same sign.

kappa=emMatrixToKappa[(({{a1, -b1, 0, 0, 0, 0}, {b1, a1, 0, 0, 0, 0}, {0, 0, a2, 0, 0, a3}, {0, 1, 0, a1, b1, 0}, {1, 0, 0, -b1, a1, 0}, {0, 0, a3, 0, 0, a2}}))];

We may assume that a3 != 0 since otherwise the Fresnel surface contains the 3-plane x0=0

fr=FullSimplify[emKappaToFresnel[kappa,{x0,x1,x2,x3}]];

FullSimplify[fr/.{x0->0,a3->0}]

Write out algebraic equations that kappa satisfies and eliminate variables for A and B

eta=kappa + mu  emIdentityKappa[];
LHS=emCompose[eta,eta];
AA=emMatrix["A",4,Structure->"AntiSymmetric"];
BB=emMatrix["B",4,Structure->"AntiSymmetric"];
RHS=-lambda emIdentityKappa[] + emBiProduct[rho, AA,BB]+emBiProduct[rho, BB,AA];

Since rho, A,B are all non-zero, we may scale A and assume that rho = 1

rho=1;

eqs=simp[Union[Flatten[LHS-RHS]]];
show[eqs]

elimVars=Join[Variables[AA],Variables[BB]]

condVars=Join[Variables[kappa],{lambda,mu}]

Eliminate variables using a Gröbner basis

gb=GroebnerBasis[eqs, condVars,elimVars];//Timing
gb=simp[gb];//Timing
Length[gb]

show[gb]

By equation (1) we have mu = -a2

subs={mu->-a2};
show[simp[gb//.subs]]

Equation (2):

   (b1^4) ((a3^2)+lambda) = 0
   
contradicts b1 != 0, a3 != 0 and lambda >0. Thus Metaclass V is not possible.