Let $\g$ be a Lie algebra. The lower central series of $\g$ is the filtration of subalgebras $$ \D_1 \g \supset \D_2 \g \supset \D_3 \g \supset \cdots \supset \D_k \g \supset \cdots $$ of $\g$ inductively defined for every natural number $k$ as follows: \begin{eqnarray*} \D_1 \g & := & [\g,\g] \\ \D_k \g & := & [\g, \D_{k-1} \g] \end{eqnarray*} The upper central series of $\g$ is the filtration $$ \D^1 \g \supset \D^2 \g \supset \D^3 \g \supset \cdots \supset \D^k \g \supset \cdots $$ defined inductively by \begin{eqnarray*} \D^1 \g & := & [\g,\g] \\ \D^k \g & := & [\D^{k-1} \g, \D^{k-1} \g] \end{eqnarray*} In fact both $\D^k \g$ and $\D_k \g$ are ideals of $\g$ and $\D^k \g \subset \D_k \g$ for all $k$ The Lie algebra $\g$ is defined to be nilpotent if $\D_k \g = 0$ for some $k \in \mathbb{N}$ and solvable if $\D^k \g = 0$ for some $k \in \mathbb{N}$

A subalgebra $\h$ of $\g$ is said to be nilpotent or solvable if $\h$ is nilpotent or solvable when considered as a Lie algebra in its own right. The terms may also be applied to ideals of $\g$ since every ideal of $\g$ is also a subalgebra.