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.

Levi subalgebras and Levi decompositions

Definition

Let $g$ be finite-dimensional with radical $r = rad (g)$ . A Levi subalgebra or Levi factor is a semisimple Lie subalgebra $s \subseteq g$ such that

$g = r \oplus s$

as vector spaces. Since $r$ is an ideal, its bracket action by $s$ makes the map $s ⋉ r \to g$ , $(x, u) \mapsto x + u$ , an isomorphism, where the acting algebra is written first as in Semidirect products of Lie algebras. This is called a Levi decomposition. This definition asserts neither existence nor uniqueness. If $g$ is semisimple, then $r = 0$ and $s = g$ is the evident Levi factor; if $g$ is solvable, a Levi factor, when it exists, must be zero.

Depends on

Dependency tree · 2 levels

Sources

Milne, Lie Algebras, Definition 6.24