Non-zero divisors over universal enveloping algebras

Question

Let $k$ be a field and $\mathfrak{g}$ a nonnegatively graded Lie $k$-algebra. Let $\xi\in\mathfrak{g}^{\mathrm{even}}\backslash\{0\}$. How do you prove that $\xi$ is a left and right nonzero divisor on the universal enveloping algebra $U\mathfrak{g}$? I can see it if $\xi$ is central but in general I don't know how to prove it.