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 $A$ on $P$. Looking to the connetion $1$-form in local sections we can write it as (as in the paper, here $A$ denotes the connection $1$-form):
$$ A=A_1 dx_1+A_2 dx_2+A_3 dx_3+A_1 dx_4 $$
And then he claims,
We now make the assumption that the Lie algebra-valued functions $A_i$ are independent of $x_3$ and $x_4$ and hence define functions of $(x_1, x_2) \in \mathbb{R}^2$. Thus $A_1$ and $A_2$ define a connection $$A=A_1 dx_1+A_2 dx_2$$ over $\mathbb{R}^2$.
Why does it define a connection over $\mathbb{R}^2$? And what is the relation of this connection with the original connection on $P$?