A PBW theorem for inclusions of (sheaves of) Lie algebroids

Inspired by the recent work of Chen-Stienon-Xu on Atiyah, classes associated to inclusions of Lie algebroids, we give a very simple criterium (in terms of those classes) for relative Poincare-Birkhoff-Witt type results to hold. The tools we use (e.g. the first infinitesimal neighbourhood Lie algebroid) are straightforward generalizations of the ones previously developped by Caldararu, Tu and the author for Lie algebra inclusions.