DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge 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, ideals, and center

Definition

Let $g$ be a Lie algebra over $k$ , and let $h$ be a linear subspace (Linear subspace of a vector space).

It 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 $g$ .
It is an ideal, written $i ⊴ g$ , if $[g, i] \subseteq i$ . Since the bracket is skew-symmetric, this is equivalent to $[i, g] \subseteq i$ .
The center is $Z (g) = {z \in g : [z, x] = 0 for every x \in g} .$

The center is an ideal: it is a linear subspace by bilinearity, and for $z \in Z (g)$ and $x \in g$ one has $[x, z] = 0$ , which lies in $Z (g)$ . Every ideal is therefore a Lie subalgebra, but a Lie subalgebra need not be an ideal.

Depends on

Used by

Not every Lie subalgebra is an ideal False statement
The quotient Lie-algebra bracket is well-defined Lemma
Derivations form a Lie algebra and inner derivations an ideal 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

Etingof, MIT 18.745 notes, §3.2 (standard reference, not scraped)
Kirillov, An Introduction to Lie Groups and Lie Algebras, §5.3 (standard reference, not scraped)