ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-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.

Abelian Lie algebras are nilpotent of class one

Example

Every nonzero abelian Lie algebra is nilpotent of class one. The zero Lie algebra has class zero under the library convention.

Facts & Assumptions

Given: An abelian Lie algebra $a$ over a field $k$ .

[L1]

Nilpotency class is the least $c \geq 0$ for which $γ_{c + 1} = 0$ , with the zero algebra assigned class zero (Nilpotency class of a Lie algebra).

Verification

technique · direct

1.1

Since $a$ is abelian, $γ_{2} (a) = [a, a] = 0$ . If $a \neq 0$ , then $γ_{1} (a) = a \neq 0$ , so the least vanishing index is $2$ and [L1] gives class one.

givenL1algebra

2.1

If $a = 0$ , already $γ_{1} (a) = 0$ , and the explicit convention in [L1] gives class zero rather than one. These are the only two cases, and the calculation uses no choice.

L1step 1.1∎

Depends on

Nilpotency class of a Lie algebra

Used by

A nilpotent-by-nilpotent extension need not be nilpotent Counterexample

Dependency tree · two levels

2 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, nilpotent Lie algebras (standard reference, not scraped)