Generalized Commutator on Octonion Algebra

Let $\mathcal O$ be an octonion algebra over a field $F$ of characteristic different from $2$, and let $A,B\in\mathcal O$, not both zero. We study the image of the generalized commutator $Φ(x,y)=A(xy)-B(yx)$ on $\mathcal O$. We prove that $Φ$ is surjective if and only if either one of $A$ and $B$ is