Derived series of a Lie algebra

Let $\mathfrak g$ be a Lie algebra. Its derived series is the descending sequence

\mathfrak g^{(0)} := \mathfrak g,\qquad \mathfrak g^{(k+1)} := [\mathfrak g^{(k)},\,\mathfrak g^{(k)}]

for $k\ge 0$ , where the bracket denotes the derived subalgebra of $\mathfrak g^{(k)}$ .

Each $\mathfrak g^{(k)}$ is a characteristic ideal of $\mathfrak g$ .

Solvability

A Lie algebra is solvable if $\mathfrak g^{(r)}=0$ for some $r\ge 0$ ; the least such $r$ is its derived length. See solvable Lie algebra.

Relation to groups

For a connected Lie group $G$ with Lie algebra $\mathfrak g$ , this is the infinitesimal analogue of repeatedly taking the commutator subgroup.