Free Lie algebra

Summary

The free Lie algebra on a set $X$ is the Lie algebra generated by $X$ with no relations other than those forced by the axioms. It is characterized by a universal property: every map from $X$ to a Lie algebra extends uniquely to a homomorphism. It is the free object in the category of Lie algebras, analogous to the free group for groups. Its elements are linear combinations of iterated brackets of elements of $X$, and bases such as the Hall basis make this precise.