# Primitive elements in Hopf algebras over the integers

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.