Let $k$ denote a field of characteristic $0$ (assume algebraically closed for convenience). Define $J=k\langle x,y|[x,y]=y^{2}\rangle$. This noncommutative algebra (which can be viewed as a derivation ring over a commutative polynomial ring) is often referred to as the "Jordan plane" in the literature.

It can be see that $J\ncong \mathcal{U}(\mathfrak{g})$ (as algebras) for any Lie algebra $\mathfrak{g}$ (since $J/\langle [J,J]\rangle $ is not semiprime). My question is as follows: does there exist a Lie algebra $\mathfrak{g}$ and ideal $I\subseteq \mathcal{U}(\mathfrak{g})$ such that $J\cong \mathcal{U}(\mathfrak{g})/I$ (as algebras)? I guess that the answer should be "no", but I'm not sure how to go about proving it. I also assume this should be well known, given the depth of literature/ knowledge about the Jordan plane.

EDIT: I'm looking for a finite dimensional Lie algebra $\mathfrak{g}$ with these properties.