I have posted this question on math.stackexchange, without success. I'll make it brief:

Let $E\rightarrow M$ be an orientable vector bundle of rank n equipped with some Riemannian metric, $P:=F_{SO(n)}(E)$ its orthonormal frame bundle. Set $P^{c}:=F_{SO(n)}(E)\times_{SO(n)} SO(n,\mathbb{C})$.

I am trying to understand why the following statement is true:

*If we have a principal connection $\omega^{c}$ on $P^{c}$, we can decompose it uniquely into $(\omega,\phi)$, with $\omega$ a principal connection on $P$ and $\phi\in \Omega^{1}_{M}(iad P)$.*

What I understand is how to decompose the connection so that $\phi\in\Omega^1_M(iadP^c)$. It should be possible in general, as long as we have a setting where $W\subset Q$ is a reduction of principal bundles s.t. the quotient of their respective lie groups form a reductive space.

I do not understand how we can reduce $\phi$ even further. Is it possible that $w^c$ needs to be flat?

Thank you very much.