I am reading the article ON THE BATCHELOR TRIVIALIZATION OF THE TANGENT SUPERMANIFOLD by O.A Sánchez Valenzuela. Right at the beginning, the following two statements appear:
1.- Let $E\rightarrow M$ be a vector bundle of rank $n$, consider the following sheaves $\mathcal{E}=\Gamma(E)$, $\mathcal{E}^{*}=\Gamma(E^{*})$, $\Lambda\mathcal{E}=\Gamma(\Lambda E)$ ($\Gamma(\cdot)$ denotes the sheaf of sections of the respective vector bundles.), and $\mathfrak{X}_{M}=Der(C^{\infty}_{M})$ (the sheaf of vector fields on $M$). Then, there exists an exact sequence: $$0\rightarrow\Lambda\mathcal{E}\otimes\mathcal{E}^{*}\rightarrow Der(\Lambda\mathcal{E})\rightarrow\Lambda\mathcal{E}\otimes\mathfrak{X}_{M}\rightarrow0.$$ The article explains how to construct the morphisms that make the sequence exact.
2.- If the vector bundle $E\rightarrow M$ has a connection $\nabla$, then the exact sequence splits, and therefore $Der(\Lambda\mathcal{E})\cong\Lambda\mathcal{E}\otimes(\mathfrak{X}_{M}\oplus\mathcal{E}^{*})$. Here the article does not explain how to construct the isomorphism between $Der(\Lambda\mathcal{E})$ and $\Lambda\mathcal{E}\otimes(\mathfrak{X}_{M}\oplus\mathcal{E}^{*})$, I have tried to do it but have not succeeded. Hopefully, someone knows how to do it or can guide me on this matter.