= Invariant gauge kinetic form
{title2=$g([X,Y],Z)+g(Y,[X,Z])=0$}
In $\mathcal L=-g_{ab}F^a_{\mu\nu}F^{b\mu\nu}/4$, only the symmetric part of $g$ contributes. A constant symmetric <bilinear form> gives local <gauge invariance> precisely when it is an <invariant bilinear form on a Lie algebra>, because the <gauge field strength> transforms in the <Adjoint representation>. A nondegenerate kinetic term requires a <nondegenerate bilinear form>; positivity of the Hamiltonian is an additional condition. For a compact simple real <Lie algebra>, $-\kappa$ is positive definite, and a positive multiple supplies the usual positive-energy convention with Minkowski signature $(+---)$.
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