invariant form (Lie algebras)

Let $V$ be a representation of a Lie algebra $𝔤$ over a field $k$. Then a bilinear form$B:V\times V\to k$ is invariant if

$$B(Xv,w)+B(v,Xw)=0.$$

for all $X\in 𝔤,v,w\in V$. This criterion seems a little odd, but in the context of Lie algebras,it makes sense. For example, the map $\tilde{B}:V\to V^{*}$ given by $v\mapsto B(\cdot ,v)$is equivariant if and only if $B$ is an invariant form.