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

Define Metaclass IV with parameters:

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

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}}))];

We may assume that a4 != 0 since otherwise the Fresnel surface contains the plane x0=x3=0

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

fr/.{x0->0,x3->0,a4->0}

We may assume that a3^2 != a4^2 since otherwise det(kappa)=0

FullSimplify[emDet[kappa]]

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]

Since lambda>0, the below equations imply that lambda = b1^2 = b2^2

show[Take[gb,{26,27}]]

subs={lambda->b1^2,b2->b1};

tmp=simp[gb//.subs];
show[Take[tmp,{16,17}]]

Since a4, b1 != 0, the below equations imply that mu = -a1 and a1 = a2

subs=Append[subs,mu->-a1]
subs=Append[subs,a2->a1]

show[simp[gb//.subs]]

subs

We have shown that 

    a4 != 0, 
    a4^2 != a3^2,
    b1 = b2
    a1 = a2. 
    
Thus the Fresnel surface decomposes into a double light cone.