Vector field and vector valued differential form - Notation Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. Let $(P,M,G,\omega)$ be a vector bundle with connection form $\omega$.
Let $X,Y$ be smooth vector fields on $P$. I am confused about the meaning of notation $X(\omega(Y))$.
I know what it means when we have a real valued differential form in place of $\omega$. 
Suppose $\omega$ is a real valued differential form then $\omega(Y)$ is a real valued smooth function on $P$ sending $p$ to $\omega(p)Y(p)$. As $X$ is a vector field, this takes a real valued smooth function to another real valued smooth function. Thus, $X(\omega(Y))$ is a real valued smooth function.
What would this mean when $\omega $ is a $\mathfrak{g}$ valued $1$ form. I guess $X(\omega(Y))$ is a $\mathfrak{g}$ valued function on $P$. 
Any reference is welcome.
This notation is used in structure equation of a connection form.

$\omega$ is a $1$-form, we deduce that for every $x\in P$, $\omega_x(Y(x))\in{\cal g}$ and $h_Y:P\rightarrow {\cal g}$ defined by $x\rightarrow\omega_x(Y(x))$, $X\omega(Y)$ is the differential of $h_Y$ evaluated at $X$, $dh_Y.X(x)$.
