Lemma 4.3: Computations for Claim 1, Case C

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

Ax = DiagonalMatrix[{a1, 0,0}]; 
Bx = emGeneralSymmetric3x3["B"];
Cx = emGeneral3x3["C"];
Dx = Transpose[Cx];

eq1=Flatten[Cx.Cx+Ax.Bx+IdentityMatrix[3]];
eq2=Flatten[Bx.Cx+Transpose[Cx].Bx];
eq3=Flatten[Cx.Ax+Ax.Transpose[Cx]];
eqs=Join[Join[eq1,eq2],eq3];

Simplify[Union[eqs]]//MatrixForm

Since a1 != 0, it follows that C11, C21, C31 = 0.

subs={};
subs=Append[subs,C11->0];
subs=Append[subs,C21->0];
subs=Append[subs,C31->0];Simplify[Union[eqs//.subs]]//MatrixForm

Equation 1+ a1 B11 =0 implies that B11 != 0
Equation 1+C23 C32+(C33^2) =0 implies that C23 != 0 and C32 != 0

Coordinate transformation

kappa=emABCDToKappa[Ax,Bx,Cx,Dx];
kappaSub=kappa//.subs;
emKappaToMatrix[kappaSub]//MatrixForm

Define Jacobian matrix for coordinate transformation

    tilde x^0 = x^0 + x^3
    tilde x^i = x^i,   i=1,2,3.
    
See emCoordinateChange in KappaLib source code.

L={
{1,0,0,1},
{0,1,0,0},
{0,0,1,0},
{0,0,0,1}
};

kappaTrans=emCoordinateChange[kappaSub,L];
{Atrans,Btrans,Ctrans,Dtrans}=emKappaToABCD[kappaTrans];
Simplify[Atrans]//MatrixForm
Simplify[Det[Atrans]]