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

## TL;DR

This paper explores the foundational PBW theorem for free Lie algebras, demonstrating its close connection to Magnus-Witt theorems through various proofs and expositions.

## Contribution

It provides new insights into the relationship between the PBW theorem and Magnus-Witt theorems, offering multiple proof strategies.

## Key findings

- PBW theorem closely follows from Magnus-Witt theorems
- Multiple proof methods elucidate the theorem's foundations
- Enhanced understanding of free Lie algebra structures

## Abstract

This expository paper focuses on free Lie $K$-algebras and the basic PBW theorem. We argue in various ways that the basic PBW theorem is a quite close consequence of the Magnus-Witt theorems concerning free Lie algebras.