Lemma 4.3: Computations for Claim 1, Case B

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

Ax = DiagonalMatrix[{a1, a2, 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[eqs]//MatrixForm

Since a1,a2 != 0, it follows that C31,C32,C11,C22 = 0.

subs={};
subs=Append[subs,C31->0];
subs=Append[subs,C32->0];
subs=Append[subs,C11->0];
subs=Append[subs,C22->0];
Simplify[Union[eqs//.subs]]//MatrixForm

Equation 1+C33^2 = 0 has no real solution for C33. 
Thus rank A can not be 2.