SetDirectory["~/writing/WIP/KappaLib/"];
<<kappaLib.m
<<helper.m

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

Factorisation of Metaclass I

kappa=emMatrixToKappa[(({{a1, 0, 0, -b1, 0, 0}, {0, a2, 0, 0, -b2, 0}, {0, 0, a3, 0, 0, -b3}, {b1, 0, 0, a1, 0, 0}, {0, b2, 0, 0, a2, 0}, {0, 0, b3, 0, 0, a3}}))];

subs={a3->a2,b3->b2};
kappa=kappa//.subs;
FullSimplify[emDet[kappa]]

fresnel=emKappaToFresnel[kappa,vars];
FullSimplify[fresnel]

DD=(((((a1-a2))^2)+(b1^2)+(b2^2))/(b1 b2));

Since a1!= a2 or b1!= b2 (or both) we have DD > 2. Hence

	(1/2) (-DD+(Sqrt[-4+(DD^2)]))<0, 		(1/2) (-DD-(Sqrt[-4+(DD^2)]))<0


and the below matrices both have Lorentz signature

AA= (({{1, 0, 0, 0}, {0, -1, 0, 0}, {0, 0, (1/2) (-DD+(Sqrt[-4+(DD^2)])), 0}, {0, 0, 0, (1/2) (-DD+(Sqrt[-4+(DD^2)]))}}));
BB = (({{1, 0, 0, 0}, {0, -1, 0, 0}, {0, 0, (1/2) (-DD-(Sqrt[-4+(DD^2)])), 0}, {0, 0, 0, (1/2) (-DD-(Sqrt[-4+(DD^2)]))}}));

Simplify[fresnel-b1 b2 b2 (vars.AA.vars)(vars.BB.vars)]

Factorisation of Metaclass II

kappa=emMatrixToKappa[(({{a1, -b1, 0, 0, 0, 0}, {b1, a1, 0, 0, 0, 0}, {0, 0, a2, 0, 0, -b2}, {0, 1, 0, a1, b1, 0}, {1, 0, 0, -b1, a1, 0}, {0, 0, b2, 0, 0, a2}}))];
subs={a2->a1,b2->b1};
kappa=kappa//.subs;
FullSimplify[emDet[kappa]]

fresnel=emKappaToFresnel[kappa,vars];
FullSimplify[fresnel]

AA=(({{1, 0, 0, b1}, {0, -b1, 0, 0}, {0, 0, -b1, 0}, {b1, 0, 0, 0}}));
BB=(({{-1, 0, 0, b1}, {0, -b1, 0, 0}, {0, 0, -b1, 0}, {b1, 0, 0, 0}}));
Simplify[fresnel-b1(vars.AA.vars) (vars.(BB).vars)]

Since 

	Det(AA) <0,
	Det(BB)<0, 

matrices AA and BB both have Lorentz signatures.

Det[AA]
Det[BB]

Factorisation of Metaclass IV

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}}))];
subs={a2->a1,b2->b1};
kappa=kappa//.subs;
FullSimplify[emDet[kappa]]

fresnel=emKappaToFresnel[kappa,vars];
FullSimplify[fresnel]

D1=(((((a1-a3))^2)-(a4^2)+(b1^2))/(a4 b1));
AA= (({{1, 0, 0, 0}, {0, (1/2) (-D1+(Sqrt[4+(D1^2)])), 0, 0}, {0, 0, (1/2) (-D1+(Sqrt[4+(D1^2)])), 0}, {0, 0, 0, -1}}));
BB= (({{1, 0, 0, 0}, {0, (1/2) (-D1-(Sqrt[4+(D1^2)])), 0, 0}, {0, 0, (1/2) (-D1-(Sqrt[4+(D1^2)])), 0}, {0, 0, 0, -1}}));
Simplify[fresnel-b1^2 a4(vars.AA.vars) (vars.(BB).vars)]

Note: We know that a4!=0 and b1!=0, so D1 is well defined. However, 
we have no inequality for D1 as in Metaclass I. Nevertheless

	
	(1/2) (-D1+(Sqrt[4+(D1^2)])) > 0,
	(1/2) (-D1-(Sqrt[4+(D1^2)]))<0.

Thus the above matrices both have Lorentz signature.