TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Solvability criterion via the derived algebra

Statement

A finite-dimensional Lie algebra $g$ over a characteristic-zero field is solvable if and only if its derived algebra $g^{'} = [g, g]$ is nilpotent.

Facts & Assumptions

Given: A finite-dimensional Lie algebra $g$ over a characteristic-zero field.

[L1]

The derived algebra of a solvable finite-dimensional characteristic-zero Lie algebra is nilpotent (The derived algebra of a solvable Lie algebra is nilpotent in characteristic zero).

[L2]

Every nilpotent Lie algebra is solvable (Nilpotent Lie algebras are solvable).

[L3]

A Lie algebra with a solvable ideal and solvable quotient is solvable (Subalgebras, quotients, and extensions of solvable Lie algebras).

Proof

technique · direct

1.1

If $g$ is solvable, [L1] says directly that $g^{'}$ is nilpotent. This is the direction that uses finite dimensionality and characteristic zero.

givenL1

2.1

Conversely suppose $g^{'}$ is nilpotent. It is solvable by [L2], while $g / g^{'}$ is abelian and hence solvable. Applying the extension assertion [L3] to the ideal $g^{'}$ shows that $g$ is solvable. This reverse direction is valid over every field and includes $g^{'} = 0$ and $g = 0$ .

L2L3algebra∎

Depends on

Used by

Cartan's solvability criterion Theorem

Dependency tree · two levels

11 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, Corollary 3.8 (standard reference, not scraped)