Lie algebras MOC

Invariant bilinear form on a Lie algebra

A bilinear form $⟨ \cdot, \cdot ⟩ : 𝔤 \times 𝔤 \to 𝕂$ on a Lie algebra $𝔤$ is said to be invariant iff the following equivalent conditions hold for any $𝑥, 𝑦, 𝑧 \in 𝔤$ ¹

$⟨ [𝑥, 𝑦], 𝑧 ⟩ = ⟨ 𝑥, [𝑦, 𝑧] ⟩$
$⟨ a d 𝑥 𝑦, 𝑧 ⟩ + ⟨ 𝑦, a d 𝑥 𝑧 ⟩ = 0$
$⟨ [𝑥, 𝑦], 𝑧 ⟩ + ⟨ 𝑦, [𝑥, 𝑧] ⟩ = 0$

See also

Quadratic Lie algebra

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1988. Vertex operator algebras and the Monster, §1.6, p. 17 ↩