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.

The two-dimensional affine Lie algebra is solvable, not nilpotent

Example

Let $a = k x \oplus k y$ with $[x, y] = y$ . Then $a^{(1)} = k y$ , $a^{(2)} = 0$ , and $γ_{r} (a) = k y$ for every $r \geq 2$ . Thus $a$ is solvable but not nilpotent.

Facts & Assumptions

Given: The displayed two-dimensional Lie algebra over a field $k$ .

[L1]

The derived series tests solvability (Derived series and solvable Lie algebras).

[L2]

The lower central series tests nilpotence (Lower central series and nilpotent Lie algebras).

Verification

technique · direct

1.1

Every basis bracket is zero or a scalar multiple of $y$ , and $[x, y] = y$ is nonzero. Hence $[a, a] = k y$ , while $[k y, k y] = 0$ . Therefore $a^{(1)} = k y$ and $a^{(2)} = 0$ , so $a$ is solvable by [L1].

givenL1algebra

1.2

The same first calculation gives $γ_{2} (a) = k y$ . Since $[a, k y] = k y$ , induction gives $γ_{r} (a) = k y \neq 0$ for every $r \geq 2$ . Thus the lower central series does not terminate and $a$ is not nilpotent by [L2].

givenL2algebra

2.1

These computations establish every displayed equality and both conclusions over every field, including characteristic two, because the persistent bracket has coefficient one. No choice is used.

step 1.1step 1.2∎

Depends on

Used by

A nilpotent-by-nilpotent extension need not be nilpotent Counterexample
Radical and nilradical of the affine Lie algebra Example

Dependency tree · two levels

4 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, two-dimensional nonabelian example (standard reference, not scraped)