{
  "abstract": "Let $\\mathbb O$ be the real octonion division algebra, let $\\mathfrak g_2=\\operatorname{Der}(\\mathbb O)$, and fix a quaternionic subalgebra $H\\subset\\mathbb O$. We ask when an element $Z$ preserving $H$ and its conjugate $TZT^{-1}$ generate $\\mathfrak g_2$, where $T$ is a prescribed involutive octonion automorphism. We prove that such a $Z$ exists if and only if $T(H)\\ne H$. Moreover, a fixed set of six derivations preserving $H$ contains a generating choice for every such $T$. For an arbitrary $Z\\in\\mathfrak g_2$, we construct a polynomial criterion with two factors: one detects reducibility on the imaginary octonions, and the other detects generated algebras of dimension at most three. An irreducible proper generated algebra must be a principal $\\mathfrak{su}(2)$. We characterize the elements admitting this case by their rotation frequencies and parametrize all corresponding involutions. For one specified involution and quaternionic subalgebra $H$, we describe the nongenerating set by four geometric families and six irredundant systems of scalar equations, and prove that it has real dimension four.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15497",
  "author_total": 3,
  "authors": [
    {
      "name": "Santiago Pineda Montoya"
    },
    {
      "name": "Johan H. Rúa Muñoz"
    },
    {
      "name": "Borut Jurčič Zlobec"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dad06fmssrpws.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15497",
  "fulltext_url": null,
  "id": "dad06fmssrpws",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15497",
  "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.GR"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2604|Algebraic and Geometric Analysis"
    ]
  ],
  "openalex": "W7213256846",
  "pdf_url": "https://arxiv.org/pdf/2609.15497",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dad06fmssrpws.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2609.15497",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15497"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Generating pairs in $\\mathfrak g_2$ related by an involution",
  "title_text": "Generating pairs in g_2 related by an involution",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/dad06fmssrpws",
  "venue": "arXiv",
  "versions": []
}