This is a quick and dirty formulation of my question, with the hope that experts will quickly figure out what I am looking for and provide a reference.
Let $G$ be a Lie group, and let $\pi:E\to G$ be a finite dimensional real or complex $G$-equivariant vector bundle, i.e., $\pi(gs)=g\pi(s)$ for all $s\in E$. Then $E$ is trivial since the simply transitive action of $G$ defines $G$-invariant sections $e_j\in C^\infty(E)$ that comprise a global frame. The same happens with the tangent bundle $TG$ and the endomorphism bundle $\mathrm{Hom}(E)$. The vector space of (left) $G$-inavriant vector fields in $C^\infty(TG)$ is a Lie algebra isomorphic to the Lie algebra $\mathrm{Lie}(G)$, whereas the vector space of $G$-invariant endomorphism fields in $C^\infty(\mathrm{Hom})$ is an associative algebra isomorphic to $\mathrm{gl}(\dim E/G)$. Let $\nabla$ be the flat connection on $E$ corresponding to $G$-translation.
Question: Let $\operatorname{D}:C^\infty(E)\to C^\infty(E)$ be a $G$-invariant partial differential operator. I believe that it can be written as $$ \operatorname{D}=\sum_{k=0}^m D_k(\nabla^k), $$ where $D_k\in\mathrm{Lie}(G)^{\otimes k}\otimes\mathrm{gl}(\dim E/G)$. I think I know how to prove this, but does anyone know of a reference where this is stated? Thank you.