{
  "abstract": "In 1910, Élie Cartan famously realized the split real form of the exceptional Lie group $G_2$ as the symmetry group of the maximally symmetric rank 2 distribution on a 5-dimensional manifold with the small growth vector (2,3,5). In this paper, we discover a new appearance of $G_2$ in the geometric theory of distributions, arising from a rank 3 distribution on an 8-dimensional manifold with the growth vector $(3,6,8)$. The algebra of infinitesimal symmetries of this distribution at any point is 29-dimensional and isomorphic to $(\\mathfrak{g}_2 \\oplus \\mathbb{R}) \\ltimes W$, where $\\mathfrak{g}_2$ is the Lie algebra of $G_2$ and $W$ is an adjoint module of $\\mathfrak{g}_2$. Our model possesses three remarkable properties. First, it is maximally symmetric among all bracket-generating rank 3 distributions with a 6-dimensional square (a family that includes both (3,6,8) and (3,6,7,8) distributions). To the best of our knowledge, this is the first example of a family of distributions defined by a set of prescribed small growth vectors in which maximal symmetry is achieved by a member whose growth vector is not the longest. Second, this model provides the first counterexample to the conjecture that all bracket-generating rank 3 distributions with a 6-dimensional square are of maximal class at a generic point (which is known to hold in dimensions 6 and 7). Third, further analysis yields the control-theoretic consequence that all abnormal extremal trajectories of this model originating at any point of the ambient manifold have a corank of at least 2. To our knowledge, this is the first example with this property among bracket-generating distributions with generic small growth vector for a given rank and ambient dimension. We also give an interpretation of our model in terms of split-octonions, more precisely, in terms of a natural algebraic structure on the tangent bundle to split octonions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.25910",
  "author_total": 3,
  "authors": [
    {
      "name": "Nicklas Day"
    },
    {
      "name": "Boris Doubrov"
    },
    {
      "name": "Igor Zelenko"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bea27g97m4jtn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.25910",
  "fulltext_url": null,
  "id": "bea27g97m4jtn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.25910",
  "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.OC"
    ],
    [
      "openalex",
      "3103|Advanced Differential Geometry Research"
    ]
  ],
  "openalex": "W7162394004",
  "pdf_url": "https://arxiv.org/pdf/2605.25910",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bea27g97m4jtn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2605.25910",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.25910"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "$G_2$ and the Maximally Symmetric (3, 8) Distribution with 6-Dimensional Square",
  "title_text": "G_2 and the Maximally Symmetric (3, 8) Distribution with 6-Dimensional Square",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/bea27g97m4jtn",
  "venue": "arXiv",
  "versions": []
}