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The first part of this article is concerned with proving the symmetric PBWtheorem using Magnus commutators. Extensions of the Dynkin--Specht--Wever lemmaand a general theorem of Nouaz\'e--Revoy type are also obtained. The secondpart focuses on free Lie $K$-algebras and the basic PBW theorem. Appendices areprovided in order make the discussion self-contained, and also putting it intocontext.

Some proofs of the Poincar\'e--Birkhoff--Witt theorem and related matters