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

Define Metaclass VII with parameters:

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

kappa=emMatrixToKappa[(({{a1, 0, 0, a4, 0, 0}, {0, a2, 0, 0, a5, 0}, {0, 0, a3, 0, 0, a6}, {a4, 0, 0, a1, 0, 0}, {0, a5, 0, 0, a2, 0}, {0, 0, a6, 0, 0, a3}}))];

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]

Equation gb[[45]]=0 contradicts lambda > 0.

show[{gb[[45]]}]

Thus kappa can not be in Metaclass VII