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.

Nilpotent transformations and nil representations

Definition

An endomorphism $T : V \to V$ is nilpotent when $T^{m} = 0$ for some positive integer $m$ , as in Nilpotent endomorphisms and their nilpotency index.

Let $ρ : g \to g l (V)$ be a representation (Representations of Lie algebras). The representation is nil if $ρ (x)$ is a nilpotent endomorphism of $V$ for every $x \in g$ . The exponent may depend on $x$ .

This condition concerns every operator in the represented Lie algebra; it is not enough to check an arbitrarily chosen vector-space basis. It is also distinct from saying that the abstract Lie algebra $g$ , or merely the image $ρ (g)$ under its bracket, is nilpotent. The zero action on any $V$ , and every action on the zero vector space, is nil.

Depends on

Used by

A nilpotent acting basis suffices for Engel's theorem False statement
Engel's common-zero-vector lemma Lemma

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

Milne, Lie Algebras, §2, Engel's theorem (standard reference, not scraped)