Let $A$ be a $\mathbb k$-algebra. If $A$ is affine commutative, by Nullstellensatz, then every prime ideal of $A$ is an intersection of maximal ideals.
To justify the notion of being primitive in https://arxiv.org/abs/2008.06605, Bell notes the following. If $A = U\mathfrak g$ is the universal enveloping algebra for some Lie algebra $\mathfrak g$, then every prime ideal of $A$ is an intersection of primitive ideals.
However, I was wondering if there is a way to see that $A$ does not satisfy the property that every prime ideal is an intersection of maximal ideals.
Equivalently, I was wondering if there is an explicit proof that $U\mathfrak g$ for a Lie algebra $\mathfrak g$ is not a Jacobson ring.
