Invariant integration by parts for a gauge covariant derivative
= Invariant integration by parts for a gauge covariant derivative
For the <Killing form> or another <invariant bilinear form on a Lie algebra>,
$$
\partial_\mu\kappa(X,Y)=\kappa(D_\mu X,Y)+\kappa(X,D_\mu Y).
$$
The connection terms cancel by invariance. Thus, for compactly supported variations, a <gauge covariant derivative> can be moved to the other factor in an integral with a minus sign, just as an ordinary derivative.
// Target: quantum-field-theory.bigb