quadratic Lie algebra

A Lie algebra $𝔤$ is said to be quadratic if $𝔤$ as a representation (under the adjoint action) admits a non-degenerate, invariant scalar product $(\cdot \mid \cdot )$ .

$𝔤$ being quadratic implies that the adjoint and co-adjoint representations of $𝔤$ are isomorphic.

Indeed, the non-degeneracy of $(\cdot \mid \cdot )$ implies that the induced map $\varphi :𝔤\to {𝔤}^{*}$ given by $\varphi (X)(Z)=(X\mid Z)$ is an isomorphism of vector spaces. Invariance of the scalar product means that$([X,Y]\mid Z)=-(Y\mid [X,Z])=(Y\mid [Z,X])$. This implies that $\varphi$ is a map of representations:

$$\varphi (a{d}_X(Y))(Z)=\varphi ([X,Y])(Z)=([X,Y]\mid Z)=(Y\mid [Z,X])=a{d}_X^{*}(\varphi (Y)(Z))$$