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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Lie subalgebras and ideals

Definition

Let $g$ be a finite-dimensional real or complex Lie algebra. A linear subspace $h \subseteq g$ is a Lie subalgebra if

$[h, h] \subseteq h .$

The restricted bracket then makes $h$ a Lie algebra: bilinearity, alternation, and Jacobi are inherited from $g$ .

A linear subspace $a \subseteq g$ is an ideal if

$[g, a] \subseteq a .$

Skew-symmetry makes this equivalent to $[a, g] \subseteq a$ . Every ideal is a Lie subalgebra. The zero subspace and $g$ are ideals, and no properness or nonzero assumption is included.

Depends on

Used by

The left-translated distribution associated to a Lie subalgebra Definition
An ideal integrates to a connected immersed normal subgroup Proposition
The Lie algebra of a Lie subgroup is a Lie subalgebra Proposition

Dependency tree · two levels

7 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

Pavel Etingof, MIT 18.745 Lie Groups and Lie Algebras I (standard reference, not scraped)
Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras (standard reference, not scraped)