= Invariant bilinear-form construction of a Yang-Mills action
If a <Lie algebra> has an invariant symmetric nondegenerate form $h$, its curvature $F$ transforms in the <Adjoint representation> and $\mathcal L=-h(F_{\mu\nu},F^{\mu\nu})/4$ is gauge invariant. Infinitesimal invariance follows from $h([\theta,X],Y)+h(X,[\theta,Y])=0$. For compact generators with $\operatorname{tr}(T^aT^b)=\delta^{ab}/2$, this gives $-\operatorname{tr}(F_{\mu\nu}F^{\mu\nu})/2=-F^a_{\mu\nu}F^{a\mu\nu}/4$. An arbitrary index contraction is not gauge invariant for an arbitrary <Lie algebra>.
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