  Recently the notion of post-Hopf algebra was introduced, with the universal
enveloping algebra of a post-Lie algebra as the fundamental example. A novel
property is that any cocommutative post-Hopf algebra gives rise to a
sub-adjacent Hopf algebra with a generalized Grossman-Larson product. By
twisting the post-Hopf product, we provide a combinatorial antipode formula for
the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie
algebra. Relating to such a sub-adjacent Hopf algebra, we also obtain a closed
inverse formula for the Oudom-Guin isomorphism in the context of post-Lie
algebras. Especially as a byproduct, we derive a cancellation-free antipode
formula for the Grossman-Larson Hopf algebra of ordered trees through a
concrete tree-grafting expression.
