I'm probably missing something obvious, but I've been wondering what the motivation is for requiring the components $A_\mu$ in a local trivialization of a gauge connection on a smooth principal $G$-bundle to lie in $\mathfrak{g}$, the Lie algebra of $G$. I can see that this gives a couple of nice properties; for example, in a local trivialization it ensures that under a gauge transformation $A'_\mu=gA_\mu g^{-1}+g\partial_\mu g$ lies in $\mathfrak{g}$, and that the curvature form $F=dA+A\wedge A$ lies in $\mathfrak{g}$ (since $\mathfrak{g}$ is closed under the Lie bracket). But is there a more intrinsic or geometric reason that $A_\mu$ must be in $\mathfrak{g}$? Thanks.