trace of $\operatorname{End}(E)$-valued $p$-form

Question

$\eta\,$ is $\,\operatorname{End}(E)$-valued $p$-form. $\;d_D\,$ is exterior covariant derivative. $\,E\,$ is Bundle.

How can I prove $$\,\operatorname{tr}(d_D\eta)=d(\operatorname{tr}(\eta))\quad ?$$

Answer 1

Proofs like this entail "unpacking" the respective definitions, (recall your earlier post: Exterior covariant derivative of $\operatorname{End}(E)$-valued $p$-form, e.g., and see also Taking trace of vector-valued differential forms) and then showing that by definition, $$\,\operatorname{tr}(d_D\eta)=d(\operatorname{tr}(\eta))$$