The symmetric Poincaré-Birkhoff-Witt theorem and Dynkin-Magnus commutators

The objective of this paper is to give alternative proofs for the symmetric Poincaré-Birkhoff-Witt theorem utilizing the Magnus recursion formulae or Dynkin’s non-commutative polynomial comparison method and simple universal algebraic principles. As an application of these principles, a theorem of Nouazé-Revoy type is also obtained.