The free Lie algebra on the set is the result of evaluating the free Lie algebra functor on object .
The subject of free Lie algebras is combinatorially rich with lots of open problems. By a 1953 theorem of A. I. Širšov (Shirshov) every Lie subalgebra of a free Lie subalgebra is free (an analogue of the Nielsen-Schreier theorem in combinatorial group theory). The study of bases of a free Lie algebra considered as a vector space is very nontrivial; special attention has been paid to so-called Hall bases.
N. Bourbaki, Lie groups and Lie algebras, Chap. II: Free Lie Algebras
Christophe Reutenauer, Free Lie algebras, Oxford Univeristy Press 1993
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C. Reutanauer, Applications of a noncommutative jacobian matrix, Journal of Pure and Applied Algebra 77, n. 2, 1992, p. 169-181, doi
wikipedia: free Lie algebra
Nantel Bergeron, Muriel Livernet, A combinatorial basis for the free Lie algebra of the labelled rooted trees, Journal of Lie Theory 20 (2010) 3–15, pdf
A. Murua, The Hopf algebra of rooted trees, free Lie algebras, and Lie series, pdf
F. Chapoton, Free pre-Lie algebras are free as Lie algebras, math.RA/0704.2153, Bulletin Canadien de Mathe’matiques