A Lie algebra $\mathfrak{g}$ is said to be quadratic if $\mathfrak{g}$ as a representation (under the adjoint action) admits a non-degenerate, invariant scalar product $(\cdot\mid\cdot)$ .
$\mathfrak{g}$ being quadratic implies that the adjoint and co-adjoint representations of $\mathfrak{g}$ are isomorphic.
Indeed, the non-degeneracy of $(\cdot\mid\cdot)$ implies that the induced map $\phi\colon\mathfrak{g}\to\mathfrak{g}^{*}$ given by $\phi(X)(Z)=(X\mid Z)$ 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 $\phi$ is a map of representations:
 $\phi(ad_{X}(Y))(Z)=\phi([X,Y])(Z)=([X,Y]\mid Z)=(Y\mid[Z,X])=ad^{*}_{X}(\phi(Y% )(Z))$