Questions about proving Poincare-Birkhoff-Witt theorem

Question

I am reading the book Introduction to Lie Algebras by Karin Erdmann and Mark J. Wildon. In a short chapter of universal enveloping Lie algebra, they shortly mentioned the PBW-basis without proving it.

I wonder is there any comprehensive and detailed proof of the theorem?

Could anyone give me some reference books or articles?

Thanks in advance!

Answer 1

This is proved in detail in Humphrey's "Introduction to Lie Algebras and Representation Theory" in section 17.4 (appropriately named Proof of PBW Theorem).