The concept of a free Lie algebra arises in the context of algebra, specifically in the study of Lie algebras. A Lie algebra is a vector space equipped with a binary operation (called the Lie bracket) that satisfies two properties: bilinearity and the Jacobi identity.
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The Lie algebra over a commutative ring generated by a set subject only to bilinearity, alternation and the Jacobi identity. Every map from to the underlying set of an -Lie algebra extends uniquely to a Lie algebra homomorphism. It can be constructed from bracketed words by quotienting by those identities; elements are represented by Lie polynomials.