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 $f = g \circ i$.