Let $(P,M,G,\pi,\cdot)$ be a principal bundle. Let $\omega$ be the connection one form for a connection $H\subset TP$. Let $X,Y$ be smooth vector fields on $P$. Then the curvature $\Omega$ of the connection form is
$\begin{align*} \Omega(X,Y)&=d\omega(hX,hY)\\ \\ &=(hX)\omega(hY)-(hY)\omega(hX)-\omega([hX,hY])\\ \\ &=-\omega([hX,hY]). \end{align*} $
What I am having trouble understanding is the second line of the above equations. As far as I am aware, $\omega$ is a $\mathfrak{g}$ valued one form, so shouldn't $\omega(hY)$ and $\omega(hX)$ be considered as either left-invariant vector fields on $G$ or equivalently some vectors in $T_e G$. Either way, I don't know how to interpret $(hX)\omega(hY)$ since $hX$ is a vector field also? It's as if two vector fields are being 'multiplied'?
Obviously I am missing something crucial here and any help would be much appreciated.
Thanks!
Since $\omega$ is a vector-valued form, the exterior derivative is taken componentwise - see e.g. here. Thus $hX \omega(hY)$ denotes the componentwise directional derivative of the $\mathfrak g$-valued function $\omega(hY)$ - i.e. after fixing a basis $e_i$ with dual basis $\theta^i$ for $\mathfrak g$, $$hX \omega(hY) = \sum_i hX (\theta^i(\omega(hY))) e_i$$ where $hX$ is now acting naturally as a directional derivative on functions.