Let $\pi: P \rightarrow M$ be a principal bundle and $s: U \rightarrow P$ be a local section where $U \subset M$ is open. Let $A$ be a connection 1-form and $F$ a curvature 2-form. Define $A_s = s^* A$ and $F_s = s^*F$ to be the local connection 1-form and local curvature 2-form respectively (i.e. the pullback of $A$ and $F$ under $s$).
Now suppose we have a chart on $U$ such that a basis for the tangent spaces above $U$ is given by $\{\partial \mu\}$. Thus in coordinates $$A_\mu = A_s(\partial_\mu)\\ F_{\mu\nu} = F_s(\partial_\mu, \partial_\nu). $$
There is a theorem that states that in these coordinates the local curvature 2-form $F_s$ can be expressed as $$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + [A_\mu, A_\nu]. \tag{1}$$ Since $A_s$ is Lie algebra-valued, $A_\mu$ is an element in the Lie algebra $\mathfrak{g}$. Here the base manifold $M$ is arbitrary, and $\partial_\mu$ is a basis element for the tangent spaces/vector fields over $U$, so how does it make sense for $\partial_\mu$ to act on $A_\nu$ as in (1)? Wouldn't $\partial_\mu$ have to be a tangent vector over the Lie algebra instead of $U$ for this to make sense?