Ad-invariance of the Killing form

Let $\mathfrak g$ be a finite-dimensional Lie algebra over a field of characteristic $0$, and let $B$ be its Killing form :

Lemma (ad-invariance).
For all $x,y,z\in\mathfrak g$,

Equivalently, $B(\mathrm{ad}_y x,z)+B(x,\mathrm{ad}_y z)=0$, i.e. each $\mathrm{ad}_y$ is skew-adjoint with respect to $B$.

Context.
Ad-invariance is the structural feature that makes the Killing form useful in studying ideals and decompositions; it is crucial in the proof that nondegeneracy of B is equivalent to semisimplicity .