The Killing form of a Lie algebra is defined as
$$ B(x,y)=\operatorname{Trace} (\operatorname{ad}_x \circ \operatorname{ad}_y) $$ I'm trying to show that, given a derivation of the Lie algebra, i.e. $\phi \in \mathfrak{gl(g)} $ such that $[\phi(x),y]+[x,\phi(y)]=0$ we have $$ B(\phi(x),y)+B(x,\phi(y))=0 $$
Thanks to the Jacobi identity i have proven the result for any $\phi \in \operatorname{ad}(\mathfrak{g})$, but i can't see how to expand the proof to all the derivations of the lie algebra.
EDIT: the question is a duplicate of Proof of an identity for the Killing form involving derivations.. I can't vote to close it because the question has no accepted answer, but the proof is in the comments