 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 $\mathrm{tr}(T^a{T}^b)={\delta }^{ab}/2$, this gives $-\mathrm{tr}(F_{\mu \nu }{F}^{\mu \nu })/2=-F_{\mu \nu }^a{F}^{a\mu \nu }/4$. An arbitrary index contraction is not gauge invariant for an arbitrary Lie algebra.