2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/66672A class of representations of a Lie superalgebra (over a commutative superring) in its symmetric algebra is studied. As an application we get a direct and natural proof of a strong form of the Poincare'-Birkhoff-Witt theorem, extending this theorem to a class of nilpotent Lie superalgebras. Other applications are presented. Our results are new already for Lie algebras.23 pages, LaTeXRepresentation Theory16S30, 16G30, 17B35 (Primary), 17B65 (Secondary)text