Let $E\rightarrow M$ be an orientable vector bundle of rank n equipped with some Riemannian metric, $P:=F_{SO(n)}(E)$ the orthonormal frame bundle. I say that $P^{c}:=F_{SO(n)}(E)\times_{SO(n)} SO(n,\mathbb{C})$.
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 $\phi\in \Omega^{1}_{M}(iad P)$
Intuitively, this should be possible by writing $so(n,\mathbb{C})=so(n) + iso(n)$ and precomposing $\omega^{c}$ with an appropriate smooth section $\iota\in Hom(TP,TP^{c})$. For $\pi_{j}$ the projection onto $so(n)$ resp. $iso(n)$, we could write $\omega_{p} = \pi_{1}(w^{c}_{p}\iota_{p})$, $\phi_{p} = i(\omega_{p} + i\pi_{2}(\omega^{c}_{p}\iota_{p}))$. But even if this turns out to be correct in its details, I cannot see how this should be unique or in any way "natural".
Thank you!