Derived series of a Lie algebra

Let $\mathfrak g$ be a Lie algebra .

Definition (derived series)

The derived series of $\mathfrak g$ is the descending sequence of Lie subalgebras

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

Each $\mathfrak g^{(k)}$ is an ideal in $\mathfrak g$ (because $[\mathfrak g,\mathfrak g^{(k+1)}]\subseteq \mathfrak g^{(k+1)}$), so the series is well behaved under quotients.

Solvability

A Lie algebra is solvable if $\mathfrak g^{(r)}=0$ for some $r\ge 0$; see solvable Lie algebra and the equivalences summarized in TFAE: solvability . The smallest such $r$ is the derived length.

Relation to groups

For a connected Lie group $G$ with Lie algebra $\mathfrak g$, the derived series is the infinitesimal analog of the derived series of the commutator subgroup . Informally: iterated commutators in the group differentiate to iterated commutators in the Lie algebra.