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

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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.

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Lower-central-series terms are characteristic ideals

Statement

For every Lie algebra $g$ and every $r \geq 1$ , the lower-central term $γ_{r} (g)$ is a characteristic ideal.

Facts & Assumptions

Given: A Lie algebra $g$ , an automorphism $f : g \to g$ , and an integer $r \geq 1$ .

[L1]

The lower central series has $γ_{1} (g) = g$ and $γ_{j + 1} (g) = [g, γ_{j} (g)]$ ; every term is an ideal (Lower central series and nilpotent Lie algebras).

[L2]

A Lie-algebra homomorphism is linear and preserves brackets (Homomorphisms of possibly infinite-dimensional Lie algebras).

Proof

technique · direct

1.1

For subspaces $A, B \subseteq g$ , bracket preservation, linearity, and surjectivity give $f ([A, B]) = [f (A), f (B)]$ .

givenL2algebra

2.1

The equality $f (γ_{1}) = g = γ_{1}$ is immediate. If $f (γ_{j}) = γ_{j}$ , then [L1] and step 1.1 give $f (γ_{j + 1}) = f ([g, γ_{j}]) = [g, γ_{j}] = γ_{j + 1}$ . Finite induction reaches the given $r$ , and [L1] supplies ideality.

L1step 1.1given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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, §2.1 (standard reference, not scraped)