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.

Lie subalgebras, ideals, and center

Definition

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