Derived subalgebra - Imported Knowlpedia Content

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 in fact an ideal; the ideal property is recorded explicitly in the lemma that $[\mathfrak g,\mathfrak g]$ is an ideal.

Basic consequences

$\mathfrak g$ is abelian iff $[\mathfrak g,\mathfrak g]=0$ .
The quotient $\mathfrak g/[\mathfrak g,\mathfrak g]$ is the abelianization of $\mathfrak g$ (a special case of quotient Lie algebra).
$\mathfrak g$ is called perfect if $[\mathfrak g,\mathfrak g]=\mathfrak g$ ; for example, any 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.