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?



The Ungraded Milnor-Moore Theorem
The "ungraded" version of the theorem -which is actually the version for the Hopf algebras- can be found in most of the classical references on the subject, although its statement and proof appears scattered among paragraphs or several different sections. For example see: 


* Sweedler's book: Hopf algebras, Theorem 8.1.5, p 176 (with no particular assumptions on the field) and section 13.1, p. 279, where the version for fields of characteristic $0$ is stated as an exercise.
* Montgomery's book: [Hopf algebras and their actions on rings](https://books.google.gr/books?id=1ePNAwAAQBAJ&printsec=frontcover&hl=el&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false), Corollary 5.6.4, (3), p.78 and Theorem 5.6.5, p. 79 (for zero characteristic),
* Abe's book: [Hopf algebras](https://books.google.gr/books?id=D0AIcewz5-8C&printsec=frontcover&hl=el&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false), in ch. 2, sect. 5.2, p. 110, theorem 2.5.3, deals with the special case of the [irreducible](https://mathoverflow.net/a/306949/85967), cocommutative hopf algebras over zero characteristic (i am not sure if the pointed case is contained in that book).


Finally, it is worth taking a look at: [A primer of Hopf algebras](http://preprints.ihes.fr/2006/M/M-06-40.pdf), sect. 3.8, theorem 3.8.2, p.45 for a more modern presentation of the topic. 


