Gauge-covariant integration by parts (${\partial }_{\mu }\kappa (Y,Z)=\kappa (D_{\mu }Y,Z)+\kappa (Y,D_{\mu }Z)$)

For Lie-algebra-valued fields and a constant invariant bilinear form on a Lie algebra, the connection terms in the displayed identity cancel. Indeed, invariance gives $\kappa ([A,Y],Z)+\kappa (Y,[A,Z])=0$. Ordinary integration by parts then yields $\int \kappa (D_{\mu }Y,Z)=-\int \kappa (Y,D_{\mu }Z)$ when boundary terms vanish, for example for compactly supported variations. This preserves covariance in variational calculations of Yang-Mills equations.