In the article, the universal enveloping algebra of a free Lie algebra on a set X is defined to be the free associative algebra generated by X.
It is said that by the Poincaré–Birkhoff–Witt theorem the universal enveloping algebra of a free Lie algebra is isomorphic to the symmetric algebra of the free Lie algebra as a graded vector space (both sides are graded by giving elements of X degree 1).
By definition, the universal enveloping algebra of a free Lie algebra can be non-commutative. The symmetric algebra of a free Lie algebra is commutative.
Are there some references which prove that the universal enveloping algebra of a free Lie algebra is isomorphic to the symmetric algebra of the free Lie algebra as a graded vector space? Thank you very much.
