Generalized Commutator on Octonion Algebra

Abstract

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 invertible and $A\neq B$, or $A$ and $B$ are nonzero zero divisors and $A-B$ is invertible. In the remaining cases, the image of $Φ$ is a hyperplane or a four-dimensional maximal totally isotropic subspace of $\mathcal O$, except when $A$ and $B$ are linearly independent zero divisors and $A-B$ is not invertible, in which case the image spans a six-dimensional subspace. Along the way, we show that every octonion of trace zero is a commutator. We also prove that the image of $Axy-Byx$ on any quaternion algebra over $F$ is a vector subspace, extending a result of Panja S., Saini P. and Singh A. (Images of polynomial maps with constants, Mathematika 71 (2025), no. 3, Paper No. e70031) for $2\times 2$ matrices over algebraically closed fields, and we give examples of other generalized polynomials whose images are not vector subspaces.