[[Lie algebras MOC]]
# Quadratic Lie algebra

A **quadratic Lie algebra** over $\mathbb{K}$ consists of a [[Lie algebra]] $\mathfrak{g}$ over $\mathbb{K}$ with a $\mathfrak{g}$-[[Invariant bilinear form on a Lie algebra|invariant]] [[Bilinear form#^symmetric|symmetric linear form]] $\langle \cdot,\cdot \rangle: \mathfrak{g } \times \mathfrak{g} \to \mathbb{K}$,[^2] #m/def/lie 
i.e. it is both a Lie algebra and a [[Quadratic space]] in a compatible way, namely
$$
\begin{align*}
\langle [x,y],z \rangle = \langle x,[y,z] \rangle 
\end{align*}
$$

  [^2]: [[Away from 2]]

## See also

- [[Category of quadratic Lie algebras]]

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#state/tidy | #lang/en | #SemBr