Given a Lie algebra $\mathfrak{g}$, we define its commutator as the vector space generated by

In general, is not true that $[\mathfrak{g},\mathfrak{g}]$ is a vector subspace. By abuse of notation often the vector space generated by $[\mathfrak{g},\mathfrak{g}]$, which is a Lie subalgebra, will be itself denoted $[\mathfrak{g},\mathfrak{g}]$, and called the derived Lie algebra or commutator subalgebra.