As an operator on smooth compactly supported sections (or, equivalently, as a densely-defined unbounded operator on $L^2(\Omega^\bullet(\operatorname{End}(E)))$ when $M$ is compact), the operator $d_{A \otimes A^*} : \Omega^1(\operatorname{End}(E)) \to \Omega^2(\operatorname{End}(E))$ has a formal adjoint $(d_{A \otimes A^*})^* : \Omega^2(\operatorname{End}(E)) \to \Omega^1(\operatorname{End}(E))$ satisfying \begin{align*} \langle d_{A \otimes A^*}(\alpha), \beta \rangle_{L^2} = \langle \alpha, (d_{A \otimes A^*})^*(\beta) \rangle_{L^2} \end{align*} for all $\alpha \in \Omega^1(\operatorname{End}(E))$ and $\beta \in \Omega^2(\operatorname{End}(E))$.