Lower central series and nilpotent Lie algebras

Let $\mathfrak{g}$ be a Lie algebra over a field (Lie algebras over a field). Its lower central series is

${\gamma }_1(\mathfrak{g})=\mathfrak{g},{\gamma }_{r+1}(\mathfrak{g})=[\mathfrak{g},{\gamma }_r(\mathfrak{g})](r\ge 1),$

where $[A,B]$ denotes the linear span of all brackets $[a,b]$ with $a\in A$ and $b\in B$. Each ${\gamma }_r(\mathfrak{g})$ is an ideal (Lie subalgebras, ideals, and center): if $I$ is an ideal, Jacobi shows $[\mathfrak{g},I]$ is an ideal, and induction starts with $I=\mathfrak{g}$. Moreover the series descends because $[\mathfrak{g},I]\subseteq I$ for an ideal $I$.

The Lie algebra $\mathfrak{g}$ is nilpotent if ${\gamma }_{c+1}(\mathfrak{g})=0$ for some integer $c\ge 0$. Thus the zero Lie algebra is nilpotent with $c=0$, while every nonzero abelian Lie algebra is nilpotent with $c=1$. No finite-dimensional or characteristic hypothesis is part of the definition.