Let $\pi \colon E \to M$ be a vector bundle with connection $\nabla$. We know from Lecture 32 that each holonomy group $\operatorname{Hol}^{\nabla}(p)$ is a Lie subgroup of $\operatorname{GL}(E_p)$. Thus its Lie algebra, denoted by $\mathfrak{hol}^{\nabla}(p)$, is a Lie subalgebra of $\mathfrak{gl}(E_p)$.

We form the holonomy algebra of $\nabla$ as the vector bundle

$$\mathfrak{hol}^{\nabla} := \bigsqcup_{p \in M} \mathfrak{hol}^{\nabla}(p).$$

In this lecture we prove that the holonomy algebra is a Lie algebra subbundle of the endomorphism bundle $\operatorname{End}(E)$.