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

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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.

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Semisimple Lie algebras are centerless and perfect

Statement

If $g$ is finite-dimensional and semisimple over a characteristic-zero field, then $Z (g) = 0$ and $[g, g] = g$ .

Facts & Assumptions

Given: Such a Lie algebra $g$ .

[L1]

Its Killing form is nondegenerate (Cartan's semisimplicity criterion).

[L2]

The Killing form is the trace form of the adjoint representation (Killing form).

Proof

technique · direct

1.1

If $z \in Z (g)$ , then ${ad}_{z} = 0$ , so [L2] gives $K (z, x) = 0$ for every $x$ . Nondegeneracy in [L1] yields $z = 0$ .

L1L2

2.1

Let $D = [g, g]$ . Its orthogonal complement consists exactly of the elements $z$ with $K ([x, y], z) = K (x, [y, z]) = 0$ for all $x, y$ , hence $[y, z] = 0$ by [L1]. Thus $D^{⊥} = Z (g) = 0$ by step 1.1, and finite-dimensional nondegeneracy gives $D = g$ .

L1step 1.1algebra

3.1

When $g = 0$ , both conclusions read $0 = 0$ ; no nonempty choice was used.

step 1.12.1∎

Depends on

Used by

Ado's theorem with nilpotent nilradical action Theorem
Derivations of semisimple Lie algebras are inner Theorem
Equivalent characterizations of reductive Lie algebras Theorem
First Whitehead lemma Theorem
Malcev conjugacy of Levi subalgebras Theorem
Weyl's complete reducibility theorem Theorem

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, consequences of Theorem 4.13 (standard reference, not scraped)