Let $P \to M$ be a principal $G$-bundle and $\langle\cdot,\cdot\rangle$ be a $G$-invariant Riemannian metric on $P$. We have an Ehresmann connection for this bundle defined by $${\rm Hor}_p(P) \doteq {\rm Ver}_p(P)^\perp,$$for any $p \in P$. So the projection $TP \to {\rm Ver}(P) \to \mathfrak{g}$ defines a connection $1$-form and so we may consider the associated curvature $2$-form $\Omega = {\rm D}\omega$, where ${\rm D}$ stands for the exterior covariant derivative given by the connection. I'd expect $\Omega$ to be somehow related to the Levi-Civita connection $\nabla$ of $\langle\cdot,\cdot\rangle$, or to the $\nabla$-horizontal connection on the vector bundle $TP\to P$.
In a very broad sense, I think my question is: how to compute $\Omega$?
I guess I'm just a bit stumped about how to concretely use the definition of ${\rm Hor}_p(P)$ as an actual orthogonal complement. I'd like to see how this particular situation is treated or, if you know a reference I can look at, I'd be happy enough (I have already skimmed through Kobayashi and Nomizu, before you suggest it -- unless I actually missed it and you have a specific page in mind). Thanks.