  We consider pairs of Lie algebras $g$ and $\bar{g}$, defined over a common
vector space, where the Lie brackets of $g$ and $\bar{g}$ are related via a
post-Lie algebra structure. The latter can be extended to the Lie enveloping
algebra $U(g)$. This permits us to define another associative product on
$U(g)$, which gives rise to a Hopf algebra isomorphism between $U(\bar{g})$ and
a new Hopf algebra assembled from $U(g)$ with the new product.
  For the free post-Lie algebra these constructions provide a refined
understanding of a fundamental Hopf algebra appearing in the theory of
numerical integration methods for differential equations on manifolds. In the
pre-Lie setting, the algebraic point of view developed here also provides a
concise way to develop Butcher's order theory for Runge--Kutta methods.
