For any Kahler manifold $M$, we have the well known Kahler 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.

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.