I have come across the following statement multiple times:
Let $G$ be a Lie group. The universal enveloping algebra of $G$ can be identified with the algebra of left-invariant differential operators acting on $C^\infty(G)$.
See here for example.
However, I cannot find a reference for this statement. (In the linked post, Helgason's Differential Geometry, Lie Groups, and Symmetric Spaces is mentioned, but I could not find anything there.)