Let $P\to M$ be a principal $G$-bundle with connection form $\omega$ and curvature form $\Omega$. Let $\operatorname{Hol}_p^0(\omega)$ denote the subgroup obtained from loops based at $\pi(p)$ that are homotopic to the constant loop. Let $P(p)$ be the set of points reachable from $p$ by piecewise smooth horizontal paths. The Lie algebra of $\operatorname{Hol}_p^0(\omega)$ is generated by the curvature elements \begin{align*} \Omega_q(X,Y), \end{align*} where $q$ ranges over $P(p)$ and $X,Y\in H_qP$.