Let $U(L)$ be the enveloping algebra of a Lie algebra $L$. How can I prove that $U(L)$ hasn't zero divisiors (e.g. if $xy=0$, $x,y \in U(L)$ then $x=0$ or $y=0$)?
The standard argument is to show that the associated graded algebra with respect to the usual filtration is a polynomial ring.
Since being an integral domain is something that goes from the associated graded algebra to the algebra, this works.
This should be done in pretty much every textbook dealing with the subject.