TITLE: characterization of subalgebras of universal enveloping algebra coming from Lie subalgebras
QUESTION [13 upvotes]: 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.

REPLY [15 votes]: In characteristic $0$, if $Y$ is a Hopf subalgebra of $U(\mathfrak{g})$, then $Y = U(\mathfrak{g}')$ for some Lie subalgebra $\mathfrak{g}'$ of $\mathfrak{g}$ (and conversely, every such subalgebra is a Hopf subalgebra). This is a consequence of Proposition 6.13 and Theorem 5.18 of John W. Milnor and John C. Moore's 1965 paper "On the structure of Hopf algebras."
If you are working over a field of positive characteristic, then the result is also true if you replace the universal enveloping algebra by the restricted enveloping algebra (see Theorem 6.11 of loc. cit.).