For any Kähler manifold $M$, we have the well known Kähler identities
\begin{align*}
[L,\partial^*] = i\overline{\partial},        & & [L,\overline{\partial}^*]=-i\partial,         & & [L,\partial] = 0,          & & [L,\overline{\partial}] = 0, \\
[\Lambda, \partial] = i\overline{\partial}^*, & & [\Lambda,\overline{\partial}] = -i\partial^*, & & [\Lambda, \partial^*] = 0, & & [\Lambda,\overline{\partial}^*] = 0,
\end{align*}
Moreover, these generalise to the analogous formuls for vector-valued forms in some holomorphic vector bundle $E$ endowed with a connection. What I would like to ask is whether the vector-valued version can be "simply" derived from the untwisted case. In the literature I have found no such derivation.