  Given a commutative algebra $A$, we exhibit a canonical structure of post-Lie
algebra on the space $A\otimes Der(A)$ where $Der(A)$ is the space of
derivations on $A$, in order to use the machinery given in [Guin & Oudom 2008]
and [Ebrahimi-Fard & Lundervold & Munthe-Kaas 2015] and to define a Hopf
algebra structure on the associated enveloping algebra with a natural action on
$A$. We apply these results to the setting of [Linares & Otto & Tempelmayr
2023], giving a simpler and more efficient construction of their action and
extending the recent work [Bruned & Katsetsiadis]. This approach gives an
optimal setting to perform explicit computations in the associated structure
group.
