In a representation theory course I took we stated and proved the following Theorem due to Cartier and Gabriel:
Theorem: Suppose $H$ is a cocommutative Hopf algebra over a field $k$ such that $ \operatorname{char} k = 0$ and $k$ is algebraically closed. Then there is an isomorphism $H \cong U \left( P (H) \right) \rtimes G(H)$ where $P(H)$ is the set of primitive elements of $H$ and $G(H)$ is the set of group-like elements.
We spent a few lectures proving this statement and then moved on to other things and I never saw any applications of this theorem. So I would like to know some nice applications of this theorem!