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!
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).