Generalized Commutator on Octonion Algebra
Abstract
Let 𝒪 be an octonion algebra over a field F of characteristic different from $2$, and let A,B∈𝒪, not both zero. We study the image of the generalized commutator Φ(x,y)=A(xy)−B(yx) on 𝒪. We prove that Φ is surjective if and only if either one of A and B is invertible and A≠ 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 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× 2 matrices over algebraically closed fields, and we give examples of other generalized polynomials whose images are not vector subspaces.
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BibTeX
@misc{mello2026generalized,
title = {{Generalized Commutator on Octonion Algebra}},
author = {Thiago Castilho De Mello and Prachi Saini},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04466},
eprint = {2610.04466},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04466},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Generalized Commutator on Octonion Algebra
AU - Thiago Castilho De Mello
AU - Prachi Saini
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04466
UR - https://arxiv.org/abs/2610.04466
AB - Let 𝒪 be an octonion algebra over a field F of characteristic different from $2$, and let A,B∈𝒪, not both zero. We study the image of the generalized commutator Φ(x,y)=A(xy)−B(yx) on 𝒪. We prove that Φ is surjective if and only if either one of A and B is invertible and A≠ 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 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× 2 matrices over algebraically closed fields, and we give examples of other generalized polynomials whose images are not vector subspaces.
ER -