Alphabeta Math
CorollaryStatement: Literature-sourcedProof: 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 derived algebra of the radical lies in the nilradical

Statement

For a finite-dimensional Lie algebra g over a characteristic-zero field,

[rad⁡(g),rad⁡(g)]⊆nilrad⁡(g).

Facts & Assumptions

Given: A finite-dimensional characteristic-zero Lie algebra g.

[L1]

The commutator of g with its radical lies in its nilradical: [g,rad⁡(g)]⊆nilrad⁡(g) (The commutator with the radical lies in the nilradical).

Proof

technique · direct
1.1

Since rad⁡(g)⊆g, monotonicity of the bracket span gives [rad⁡(g),rad⁡(g)]⊆[g,rad⁡(g)].

givenalgebra
2.1

Combining step 1.1 with [L1] gives the desired containment. This also covers zero radical, solvable g, and g=0, and uses no choice.

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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