DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

How statement and proof provenance work

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Solvable length of a Lie algebra

Definition

If $g$ is solvable, its derived length (or solvable length) is

$dl (g) = \min {m \geq 0 : g^{(m)} = 0},$

where $g^{(m)}$ is the derived series from Derived series and solvable Lie algebras. The minimum exists precisely because solvability makes the displayed subset of $N$ nonempty.

With this indexing, $dl (0) = 0$ , while every nonzero abelian Lie algebra has derived length $1$ . We leave the derived length undefined for a nonsolvable Lie algebra rather than assigning it the symbol $\infty$ .

Depends on

Derived series and solvable Lie algebras

Used by

The plane Euclidean-motion Lie algebra is solvable Example

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Milne, Lie Algebras, §3.2 (standard reference, not scraped)