SetDirectory["~/KappaLib/"];
<<kappaLib-1.1.m
<<helper.m

Metaclass III:

vars={x0,x1,x2,x3};

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

fr=emKappaToFresnel[kappa,vars];
Collect[FullSimplify[Expand[fr]],vars]

We assume that the Fresnel polynomial factorises:

A=Table[ToExpression["A"<>ToString[Min[{i,j}]]<>ToString[Max[{i,j}]]],{i,0,3},{j,0,3}];
B=Table[ToExpression["B"<>ToString[Min[{i,j}]]<>ToString[Max[{i,j}]]],{i,0,3},{j,0,3}];
A//MatrixForm
B//MatrixForm
factorised=(vars.A.vars) (vars.B.vars);

Compute Gröbner basis constraints on A and B

cons=Union[Flatten[CoefficientList[fr-factorised,vars]]];

gb=GroebnerBasis[cons,Variables[cons]];//Timing

Length[gb]
Take[emSort[gb],5]//show

Equation (1) contradicts b1 != 0.