I'm reading Hitchin's paper Self-duality Equations on a Riemann Surface (Hitchin, self duality)
In the first chapter on pages 63/64 he considers a principal $G$-bundle $P$ over $\mathbb R^4$
and a connection $A\in\Omega^1(\mathbb R^4,ad(P))$ on $P$ for some Lie group $G$ with Lie algebra $\mathfrak g$.
Locally one can write the curvature $F\in\Omega^2(\mathbb R^4,ad(P))$ of $A$ as
$$ F = \sum_{i<j} F_{ij} dx_i\wedge dx_j$$
where we have $F_{ij}\colon U\subseteq\mathbb R^4\rightarrow \mathfrak g$, so
$F$ is a Lie algebra valued two form.
Similar we have
$$A = A_1 dx_1 + A_2 dx_2 + A_3 dx_3 + A_4 dx_4$$
with $A_i\colon U \rightarrow \mathfrak g$.
After that he wrote that one can write $$\tag{$\star$ }F_{ij} = [\nabla_i,\nabla_j]$$ where $$\tag{$\star\star$}\nabla_i = \frac{\partial}{\partial x_i} +A_i \quad(\ast\ast)$$ should be a covariant derivative.
At this point I don't know how to interprete this equalities since I don't know how to combine the $A_i$ and the $\frac{\partial}{\partial x_i}$ which are form different spaces.
Questions:
Did I misinterprete the functions $A_i$ and $F_{ij}$?
If not, how does the equation $(\star\star)$ make sense?
Which Lie bracket is used in $(\star)$?