Universal enveloping algebra of the Lie algebra of vector fields on a manifold

Question

Suppose we have a smooth manifold $M$ with structure sheaf $\mathcal{O}_M$ and $\mathfrak{X}(M)$ is the Lie algebra of vector fields on $M$. How should we interpret the universal enveloping algebra $U(\mathfrak{X}(M))$? Is it just the algebra of all (finite order) differential operators $\mathcal{O}_M \to \mathcal{O}_M$?

Answer 1

It isn't, because it does not contain the operators given by multiplication by a non-constant function.

You can view the Lie algebra as (part of) a Lie-Rinehart algebra, and then the corresponding enveloping algebra is indeed that of finite order differential operators.