I'm reading Akhil Mathew's blog post on Formal Lie Theory in Characteristic Zero.

Let $H$ be cocommutative Hopf algebra over a field $k$. We can form $\mathfrak{g}$, the Lie algebra over $k$ consisting of the primitive elements of $H$. There is a canonical map $\phi : U(\mathfrak{g}) \rightarrow H$ from the universal enveloping algebra of $\mathfrak{g}$ to $H$. In fact, the functor $P : \textbf{Hopf} \rightarrow \textbf{Lie}$ sending a Hopf algebra over $k$ to its subspace of primitive elements (which forms a Lie algebra over $k$) is right adjoint to the universal enveloping algebra functor $U : \textbf{Lie} \rightarrow \textbf{Hopf}$, under certain finiteness conditions on both categories.

I am wondering,

(a) Under what conditions is $\phi$ (as above) surjective? Of course, if $H$ is generated by its primitive elements then this is true.

(b) When will $\text{im}(\phi)$ contain the group-like elements of $H$?