DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Killing form

Definition

For a finite-dimensional Lie algebra $g$ , its Killing form is the trace form of the adjoint representation:

$K_{g} (x, y) = {tr}_{g} ({ad}_{x} {ad}_{y}) .$

Here ${ad}_{x} (z) = [x, z]$ . Thus the trace is taken on the vector space $g$ itself. In particular, if $g = 0$ or if $g$ is abelian, then every adjoint operator is zero and $K_{g} = 0$ . Symmetry and invariance will follow from the general trace-form calculation rather than being included in the definition.

Depends on

Used by

Semisimple Lie algebras are centerless and perfect Corollary
A reductive algebra with degenerate Killing form Example
Classical simple Lie algebras and their Killing forms Example
Killing form of sl₂ Example
The Killing form is nondegenerate on every reductive Lie algebra False statement
Cartan's solvability criterion 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, §4, Killing form (standard reference, not scraped)