Let $H$ be a Hopf algebra over $mathbb Z$, and assume that $H$ is cocommutative, graded, generated in degree $1$, and connected (its degree-$0$ part is $mathbb Z$).
Are there nice, natural conditions that will enforce that $H$ is a universal enveloping algebra of a Lie algebra over $mathbb Z$?
For example, if $H$ is $mathbb Z$-free, then the Milnor-Moore theorem implies $Hotimesmathbb Q=U(P)$ for $P’$ the space of primitives in $Hotimesmathbb Q$, and presumably $P’=Potimesmathbb Q$ for $P$ the $mathbb Z$-module of primitives in $H$.
I’m sure this works in a much more general setting, but I failed to locate relevant papers or books on Hopf algebras over non-fields.
Note that this question is related to the MO question
Integral Milnor-Moore theorem, though it seems orthogonal.