Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Let and $P$ a $G$bundle with a Riemannian base space $(M,g)$. Denote $\mathfrak{h}$ the Lie subalgebra formed by the Killing vector fields within the vector fields of $M$. This subalgebra acts on the space of vector fields via the adjoint representation. I would like to know if there is natural representation of $\mathfrak{h}$ over the space $\Omega^1(M, \mathfrak{g})$ of Lie algebra valued forms.
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Sure: the entire Lie algebra of vector fields acts on forms and so vectorvalued forms and, in particular, $\mathfrak{g}$valued forms. Now restrict this action to $\mathfrak{h}$. However, yr principal bundle $P$ plays absolutely no role at all in this so I suspect I am not answering the question you really meant to ask. 

