Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{g}'$ its subalgebra. Then the universal enveloping algebra $U(\mathfrak{g}')$ can be canonically embedded into $U(\mathfrak{g})$, that of $\mathfrak{g}$.

Now I'm interested in the reverse direction. Given a subalgebra $Y$ of $U(\mathfrak{g})$, under what conditions on $Y$, $Y$ is the universal enveloping algebra $U(\mathfrak{g}')$ of some Lie subalgebra $\mathfrak{g}'$ of $\mathfrak{g}$?

By considering the center of $U(\mathfrak{g})$, not all subalgebras of $U(\mathfrak{g})$ arise as universal enveloping algebras of Lie subalgebras.

I searched the internet but couldn't find anything related to this question. Much appreciated for any answer or reference.