{
  "abstract": "We review some topics in \"exceptional mathematics'' from the perspective of 3-dimensional geometry: the octonions $\\mathbb{O}$, the split octonions $\\mathbb{O}'$, the bioctonions $\\mathbb{O}_\\mathbb{C} \\cong \\mathbb{C} \\otimes_\\mathbb{R} \\mathbb{O}$, the complex Albert algebra $\\mathfrak{h}_3(\\mathbb{O}_\\mathbb{C})$, and the complex form of the exceptional Lie algebra $\\mathfrak{e}_6$. We show how to functorially build an algebra isomorphic to $\\mathbb{O}$ from any 3d complex vector space equipped with an inner product and complex volume form. Similarly, we build one isomorphic to $\\mathbb{O}'$ starting from a 3d real vector space equipped with a volume form, and one isomorphic to $\\mathbb{O}_\\mathbb{C}$ starting from a 3d complex vector space equipped with a complex volume form. We give applications to 3-dimensional real and complex manifolds. Finally, we describe how to build an Jordan algebra isomorphic to $\\mathfrak{h}_3(\\mathbb{O}_\\mathbb{C})$ starting from three 3d complex vector spaces equipped with complex volume forms. This last construction gives a nice explicit description of the complex Lie algebra $\\mathfrak{e}_6$ and its subalgebra $\\mathfrak{sl}(3,\\mathbb{C}) \\oplus \\mathfrak{sl}(3,\\mathbb{C}) \\oplus \\mathfrak{sl}(3,\\mathbb{C})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07538",
  "author_total": 2,
  "authors": [
    {
      "name": "John C. Baez"
    },
    {
      "name": "Susanne Pumpluen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/920xsq3a1bhnp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07538",
  "fulltext_url": null,
  "id": "920xsq3a1bhnp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07538",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2604|Algebraic and Geometric Analysis"
    ]
  ],
  "openalex": "W7212051847",
  "pdf_url": "https://arxiv.org/pdf/2609.07538",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/920xsq3a1bhnp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07538",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07538"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Three-Dimensional Geometry in Exceptional Algebra",
  "title_text": "Three-Dimensional Geometry in Exceptional Algebra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/920xsq3a1bhnp",
  "venue": "arXiv",
  "versions": []
}