{
  "abstract": "We continue the study of the octonionic spectrum of the octonionic structure operator $h$ on $W=\\mathbb O\\otimes_{\\mathbb R}V$, where $V$ is the seven-dimensional representation of $G_2$. The real spectrum and the octonionic one-step spectrum, given by the equation $h(w)=wλ$ for real or octonion $λ$, have been determined in earlier work. In this paper, we study the two-step equation $h(w)=(wλ_1)λ_2$. Due to non-associativity, this does not in general reduce to the one-step problem. An adapted quaternionic frame represents successive right multiplication by an operator with four real coefficients. The $SU(2)$ subgroup fixing the frame splits $W$ into invariant sectors of dimensions $16$, $12$ and $28$. We factor the determinant of the resulting 4-parameter operator pencil and characterize the pairs for which a nonzero solution exists. The singular locus of this pencil in $\\mathbb R^4$ is the union of an explicit algebraic set and a round $3$-sphere. The $12$-dimensional sector carries the one-step problem at the reversed product $λ_2λ_1$. The $28$-dimensional sector contributes no additional real singular points; the fixed sector produces solutions even when neither product lies in the one-step right spectrum.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26613",
  "author_total": 1,
  "authors": [
    {
      "name": "Sergey Grigorian"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/90a0bq7aztfr3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26613",
  "fulltext_url": null,
  "id": "90a0bq7aztfr3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26613",
  "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.DG"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2604|Algebraic and Geometric Analysis"
    ]
  ],
  "openalex": "W7223026457",
  "pdf_url": "https://arxiv.org/pdf/2610.26613",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/90a0bq7aztfr3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.26613",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26613"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The two-step spectral locus of the octonionic structure operator",
  "title_text": "The two-step spectral locus of the octonionic structure operator",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/90a0bq7aztfr3",
  "venue": "arXiv",
  "versions": []
}