Zero-product property for enveloping algebras

Question

Let $L$ be a finite-dimensional Lie algebra $L$ over a field $k$.

Let $(U(L), i)$ be a universal enveloping algebra of $L$.

If $x,y \in U(L) - \{0\}$ is there something contradictory about the following statement? $$yzx = 0 \ \ \forall z\in U(L)$$

Answer 1

The universal enveloping algebra $U(L)$ of a finite-dimensional Lie algebra is a Noetherian integral domain. This is a well-known fact, and the proof can be found in many books on Lie algebras.
The idea is as follows. The associative $k$-algebra $U(L)$ has an associated graded ring ${\rm gr} (U(L))$, defined by a standard filtration. This graded ring is a commutative $k$-algebra, and in fact a polynomial ring over $k$ by the Poincare-Birkhoff-Witt theorem (PBW-theorem). In particular ${\rm gr} (U(L))$ is an integral domain, hence $U(L)$ is an integral domain itself (this follows directly from the standard filtration). This means, by definition, that $U(L)$ has no zero divisors.