Let $\beta $ be a bilinear form in a $K$-vector space $V$. I am trying to show that the subspace $$\mathfrak{gl} (V, \beta )=\{\phi \in \mathfrak{gl}(V) : \beta (\phi x, y)+\beta (x, \phi y)=0 \ \ \forall x, y \in V \}$$ is in fact a Lie subalgebra. It all boils down basically to showing that if $\phi, \psi \in \mathfrak{gl}(V, \beta )$ then $$(\beta (\phi \psi x, y )+\beta (x, \phi \psi y))-(\beta (\psi \phi x, y )+\beta (x, \psi \phi y))=0 $$ but this seems to require showing that $\phi \psi \in \mathfrak{gl}(V, \beta )$ but this doesn’t seem obvious if it is true at all.
Any pointers?