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

In this notebook we show that the a_i and b_i constants in Metaclass I can be ordered into any order 
by a change of coordinates

Permutate b1 <---> b2,    a1 <---> a2.

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}}))];

(* Define coordinate change. *)
jacobian=(({{1, 0, 0, 0}, {0, 0, -1, 0}, {0, 1, 0, 0}, {0, 0, 0, 1}}));
(* Note: change is orientation preserving. *)
Det[jacobian]

kappaT=emCoordinateChange[kappa,jacobian];
emKappaToMatrix[kappaT]//MatrixForm

Permutate b2 <---> b3,    a2 <---> a3.

(* Define coordinate change. *)
jacobian=(({{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 0, -1}, {0, 0, 1, 0}}));
(* Note: change is orientation preserving. *)
Det[jacobian]

kappaT=emCoordinateChange[kappa,jacobian];
emKappaToMatrix[kappaT]//MatrixForm