Let $k$ be a field of characteristic $0$.

There is a functor $U$ from Lie-algebras over $k$ to Hopf algebras over $k$ sending a $k$-Lie algebra $\mathfrak{g}$ to its universal enveloping algebra $U(\mathfrak{g})$, and a functor $P$ from the category of Hopf algebras over $k$ to the category of $k$-Lie algebras sending a Hopf algebra $H$ over $k$ to the set $P(H)$ of its primitive elements, which acquires the structure of a Lie algebra. There is an adjoint relationship where $U \dashv P$.

We may ask, "for which $H$ is the counit $\epsilon_H : U(P(H)) \rightarrow H$ of this adjunction an isomorphism". The Milnor-Moore theorem says that $\epsilon_H$ is an isomorphism when $H$ is graded, co-commutative, and generated by its primitive elements.

But, for my purposes, $H$ is not a graded Hopf algebra, merely co-commutative and generated by its primitive elements. I have only been able to find expositions of the Milnor-Moore theorem which use graded Hopf algebras and graded Lie-algebras, but I am interested in the general case.

For instance, I have seen the paper "On the structure of Hopf algebras", by Milnor and Moore, but this seems to work only in the graded case from what I can see.

Could someone direct me to the most accessible exposition (that they know of) of an ungraded version of the Milnor-Moore theorem?

Lie groups and Lie algebras, §1, Theorem 1. $\endgroup$ – abx Aug 7 '18 at 19:56