  We prove that the category of cocommutative Hopf algebras over a field is a
semi-abelian category. This result extends a previous special case of it, based
on the Milnor-Moore theorem, where the field was assumed to have zero
characteristic. Takeuchi's theorem asserting that the category of commutative
and cocommutative Hopf algebras over a field is abelian immediately follows
from this new observation. We also prove that the category of cocommutative
Hopf algebras over a field is action representable. We make some new
observations concerning the categorical commutator of normal Hopf subalgebras,
and this leads to the proof that two definitions of crossed modules of
cocommutative Hopf algebras are equivalent in this context.
