ExampleConstruction: Literature-sourcedVerification: 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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A reductive algebra with degenerate Killing form

Example

For $n \geq 1$ over a characteristic-zero field, ${g l}_{n}$ is reductive but its Killing form is degenerate:

${g l}_{n} = k I \oplus {s l}_{n}, K (I, X) = 0 (X \in {g l}_{n}) .$

Facts & Assumptions

Given: The matrix Lie algebra ${g l}_{n} (k)$ , $n \geq 1$ , over a characteristic-zero field.

[L1]

A finite-dimensional Lie algebra is reductive when it is the direct sum of its center and a semisimple ideal (Equivalent characterizations of reductive Lie algebras).

[L2]

Its Killing form is $K (X, Y) = tr ({ad}_{X} {ad}_{Y})$ (Killing form).

Verification

technique · direct

1.1

Since $n$ is invertible in $k$ , every matrix has the unique decomposition $X = \frac{tr X}{n} I + (X - \frac{tr X}{n} I),$ whose second term is traceless. Thus ${g l}_{n} = k I \oplus {s l}_{n}$ . The first summand is the center and the second is semisimple for $n \geq 2$ ; for $n = 1$ it is zero, which is semisimple by convention. Hence [L1] makes ${g l}_{n}$ reductive for every $n \geq 1$ .

L1givenalgebra

2.1

Since $I$ is central, ${ad}_{I} = 0$ . Therefore [L2] gives $K (I, X) = 0$ for every $X$ . The nonzero vector $I$ lies in the radical of the form, so it is degenerate; at $n = 1$ it is identically zero. The calculation is finite and uses no choice.

L2step 1.1algebra∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Kirillov, An Introduction to Lie Groups and Lie Algebras, Theorem 5.49 (standard reference, not scraped)