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

Verify that the Fresnel polynomial for the SDCM medium class in 

   Lindell, Bergamin, Favaro : Decomposable medium conditions in four - dimensional representation
   
factorises into the product of two quadratic forms.

Define sigma as the most general (2,2)-tensor with only a skewon part

(* 
   Any kappa with only a skewon part can be written as

    S /\ Id + Id /\ S

 for a trace-free (1,1)-tensor S and the identity (1,1)-tensor Id.

See PDCM notebook, and for the emPQToKappa routine, see the KappaLib source code 
*)
S=emMatrix["sh",4,Structure->"General"];
S=S-1/4Tr[S]IdentityMatrix[4];
id=IdentityMatrix[4];
sigma=Simplify[emPQToKappa[id,S]+emPQToKappa[S,id]];

(* Verify: medium has only a skewon part *)
Union[Flatten[Simplify[sigma+emPoincare[sigma]]]]

Other parameters in SDCM medium

AA=emMatrix["a",4,Structure->"AntiSymmetric"];
BB=emMatrix["b",4,Structure->"AntiSymmetric"];
(* rho = scalar density of weight 1 *)
(* C1 = constant *)

Define kappa as most general SDCM medium

kappa=C1 emIdentityKappa[] + emBiProduct[rho,AA,BB] + emBiProduct[rho,BB,AA] + sigma;

Check that Fresnel polynomial factorises into product of two quadratic forms

vars={x0,x1,x2,x3};
fresnel=emKappaToFresnel[kappa,vars];
fresnelExp=4rho  (vars.S.AA.vars)( vars.S.BB.vars);
Simplify[fresnel-fresnelExp]