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

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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.

Used by

The quadratic Casimir element Definition
The Killing form is invariant and nondegenerate on a complex semisimple Lie algebra Proposition
The Killing form pairs only opposite root spaces Proposition

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Pavel Etingof, Lie Groups and Lie Algebras I (standard reference, not scraped)