SetDirectory["~/KappaLib"];
<<kappaLib-1.1.m

Define transformation matrices

J1=DiagonalMatrix[{-1,1,1,1}];
J2=DiagonalMatrix[{1,1,-1,1}];

Det[J1]
Det[J2]

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}}))];

emKappaToMatrix[emCoordinateChange[kappa,J1]]//MatrixForm

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}}))];
FullSimplify[emKappaToMatrix[emCoordinateChange[kappa,J2]]]//MatrixForm

Metaclass III

kappa=emMatrixToKappa[(({{a1, -b1, 0, 0, 0, 0}, {b1, a1, 0, 0, 0, 0}, {1, 0, a1, 0, 0, -b1}, {0, 0, 0, a1, b1, 1}, {0, 0, 1, -b1, a1, 0}, {0, 1, b1, 0, 0, a1}}))];
emKappaToMatrix[emCoordinateChange[kappa,J2]]//MatrixForm

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}}))];
emKappaToMatrix[emCoordinateChange[kappa,J1]]//MatrixForm

Metaclass V

kappa=emMatrixToKappa[(({{a1, -b1, 0, 0, 0, 0}, {b1, a1, 0, 0, 0, 0}, {0, 0, a2, 0, 0, a3}, {0, 1, 0, a1, b1, 0}, {1, 0, 0, -b1, a1, 0}, {0, 0, a3, 0, 0, a2}}))];
emKappaToMatrix[emCoordinateChange[kappa,J2]]//MatrixForm

Metaclass VI

kappa=emMatrixToKappa[(({{a1, 0, 0, -b1, 0, 0}, {0, a2, 0, 0, a4, 0}, {0, 0, a3, 0, 0, a5}, {b1, 0, 0, a1, 0, 0}, {0, a4, 0, 0, a2, 0}, {0, 0, a5, 0, 0, a3}}))];
emKappaToMatrix[emCoordinateChange[kappa,J1]]//MatrixForm

Metaclass VII

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}}))];
emKappaToMatrix[emCoordinateChange[kappa,J1]]//MatrixForm