{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04466",
  "author_total": 2,
  "authors": [
    {
      "name": "Thiago Castilho De Mello"
    },
    {
      "name": "Prachi Saini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c5dv94r2deb2r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04466",
  "fulltext_url": null,
  "id": "c5dv94r2deb2r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04466",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2604|Algebraic and Geometric Analysis"
    ]
  ],
  "openalex": "W7220717954",
  "pdf_url": "https://arxiv.org/pdf/2610.04466",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c5dv94r2deb2r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04466",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04466"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Generalized Commutator on Octonion Algebra",
  "title_text": "Generalized Commutator on Octonion Algebra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/c5dv94r2deb2r",
  "venue": "arXiv",
  "versions": []
}