Derived subalgebra - Knowlpedia

Let $\mathfrak g$ be a Lie algebra.

The derived subalgebra (or commutator subalgebra) of $\mathfrak g$ is

[\mathfrak g,\mathfrak g] := \mathrm{span}\{[x,y] : x,y\in \mathfrak g\}\subseteq \mathfrak g.

It is a Lie subalgebra and an ideal.

Basic consequences

$\mathfrak g$ is abelian if and only if $[\mathfrak g,\mathfrak g]=0$ .
The quotient $\mathfrak g/[\mathfrak g,\mathfrak g]$ is the abelianization of $\mathfrak g$ .
The algebra $\mathfrak g$ is perfect if $[\mathfrak g,\mathfrak g]=\mathfrak g$ ; every nonabelian simple Lie algebra is perfect.

Remarks

The derived subalgebra is the first step in the derived series, which detects solvability and organizes many structure theorems such as the Levi decomposition.