[[Lie algebras MOC]]
# Invariant bilinear form on a Lie algebra

A [[bilinear form]] $\langle \cdot,\cdot \rangle : \mathfrak{g} \times \mathfrak{ g} \to \mathbb{K}$ on a [[Lie algebra]] $\mathfrak{g}$ is said to be **invariant** iff the following equivalent conditions hold for any $x,y,z \in \mathfrak{g}$[^1988]

- $\langle [x,y],z \rangle = \langle x,[y,z] \rangle$
- $\langle \ad_{x} y, z \rangle + \langle y, \ad_{x}z \rangle = 0$
- $\langle [x,y], z \rangle + \langle y,[x,z] \rangle = 0$

  [^1988]: 1988\. [[Sources/@frenkelVertexOperatorAlgebras1988|Vertex operator algebras and the Monster]], §1.6, p. 17

## See also

- [[Quadratic Lie algebra]]

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