Proposition 2

Subalgebras, quotient algebras and central extensions of a nilpotent Lie algebra are nilpotent. A finite product of nilpotent Lie algebras is a nilpotent Lie algebra. Let $g$ be a Lie algebra, $g'$ a subalgebra of $g$, $\mathfrak{h}$ an ideal of $g$, $\mathfrak{k} = g/\mathfrak{h}$ and $\phi$ the canonical mapping of $g$ onto $\mathfrak{k}$. If $g$ is nilpotent, then $\mathscr{C}^k g = \{0\}$ for some integer $k$, hence $\mathscr{C}^k g' \subset \mathscr{C}^k g = \{0\}$ and $\mathscr{C}^k \mathfrak{k} = \phi(\mathscr{C}^k g) = \{0\}$ and hence $g'$ and $\mathfrak{k}$ are nilpotent. If $\mathfrak{k}$ is nilpotent and $\mathfrak{h}$ is contained in the centre of $g$, then $\mathscr{C}^k \mathfrak{k} = \{0\}$ for some integer $k$, hence $\mathscr{C}^k g \subset \mathfrak{h}$ and therefore $\mathscr{C}^{k+1} g \subset [\mathfrak{h}, g] = \{0\}$, so that $g$ is nilpotent. Finally, the assertion concerning products follows for example from the assertion (a) $\Leftrightarrow$ (d) of Proposition 1.

Definition 1 and Proposition 2 show that nilpotent Lie algebras are precisely the algebras obtained from commutative Lie algebras by a sequence of central extensions.