A free Lie algebra, over a given field $K$, is a Lie algebra generated by a set $X$, without any imposed relations other than the defining relations of alternating bilinearity and the Jacobi identity.
Let $$X$$ be a set and $$i: X \to L$$ a morphism of sets from $$X$$ into a Lie algebra $$L$$. The Lie algebra $$L$$ is called free on $$X$$ if for any Lie algebra $$A$$ with a morphism of sets $$f: X \to A$$, there is a unique Lie algebra morphism $$g: L \to A$$ such that $$f = g \circ i$$.