Given a commutative algebra A, we exhibit a canonical structure of post-Lie algebra on the space A⊗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.
