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

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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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Distinct conjugate Levi subalgebras

Example

Let $V$ be a finite-dimensional ${s l}_{2}$ -module whose action is nontrivial, and set $g = {s l}_{2} ⋉ V$ . For a suitable $v \in V$ , the standard factor $s = {s l}_{2} \oplus 0$ and $\exp ({ad}_{(0, v)}) (s)$ are distinct Levi subalgebras, conjugate by the inner unipotent automorphism $\exp ({ad}_{(0, v)})$ .

Facts & Assumptions

Given: A finite-dimensional module with nonzero action and the displayed semidirect product over a characteristic-zero field.

[L1]

The semidirect-product bracket is $[(x, v), (y, w)] = ([x, y], x w - y v)$ (Semidirect products of Lie algebras).

[L2]

Levi factors are conjugate by finite products of automorphisms $\exp (ad n)$ with $n$ in the nilradical (Malcev conjugacy of Levi subalgebras).

Verification

technique · explicit conjugation

1.1

Since the action is nonzero, choose $v \in V$ and $x \in {s l}_{2}$ with $x v \neq 0$ . By [L1], $[(0, v), (y, w)] = (0, - y v)$ lies in $0 \oplus V$ , and $[0 \oplus V, 0 \oplus V] = 0$ . Therefore $({ad}_{(0, v)})^{2} = 0$ and $\exp ({ad}_{(0, v)}) = 1 + {ad}_{(0, v)}$ .

givenL1algebra

2.1

In particular, $\exp ({ad}_{(0, v)}) (x, 0) = (x, - x v) .$ Its second component is nonzero for the chosen pair, so the image subalgebra is not $s$ . An automorphism carries a Levi factor to a Levi factor, so both are Levi subalgebras.

step 1.1algebra

3.1

This explicit automorphism is a one-factor instance of the finite products in [L2], with $(0, v)$ in the abelian nilpotent ideal $0 \oplus V$ . Thus the example witnesses both literal nonuniqueness and Malcev conjugacy. Selecting one pair from “the action is nonzero” uses no choice family.

L2step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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, Malcev conjugacy (standard reference, not scraped)