Free Lie algebra on a vector space

Let $V$ be a complex vector space and let $T(V)={⨁}_{n\ge 0}{V}^{\otimes n}$ be its tensor algebra (Tensor algebra of a vector space). The commutator bracket $[x,y]=xy-yx$ makes $T(V)$ a Lie algebra whose underlying vector space is $T(V)$. The free Lie algebra on $V$, written $L(V)$, is the Lie subalgebra of $T(V)$ (Lie subalgebras, ideals, and center) generated by the image of $V=V^{\otimes 1}$, that is, the smallest Lie subalgebra of $T(V)$ containing $V$. Elements of $L(V)$ are finite linear combinations of iterated commutators of elements of $V$.

The terminology "free" refers to the universal property proved in Universal property of the free Lie algebra: every linear map from $V$ to a complex Lie algebra extends uniquely to a homomorphism of Lie algebras from $L(V)$. The construction is licensed by the Poincaré-Birkhoff-Witt theorem, which identifies $T(V)$ with the universal enveloping algebra of the free Lie algebra and shows in particular that $V$ embeds in $L(V)$ and that $L(V)=0$ when $V=0$, $L(V)=V$ when $\dim V=1$, and $L(V)$ is infinite-dimensional when $\dim V\ge 2$.