DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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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.
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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The Killing form of a semisimple Lie algebra

Definition

Let $g$ be a complex semisimple Lie algebra. For $x, y \in g$ , let ${ad}_{x}, {ad}_{y} \in End (g)$ be the adjoint operators. The Killing form of $g$ is the symmetric bilinear form

$B (x, y) : = tr ({ad}_{x} {ad}_{y}) .$

The trace is taken on the finite-dimensional vector space $g$ . This is the invariant form used to define dual bases, the root-theoretic vectors $H_{α}$ , and the quadratic Casimir element on this page.

The Killing form of a semisimple Lie algebra

Definition

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