{
  "abstract": "We study the Coxeter-Dickson $E_8$ order as a common integral support for the octonionic, para-octonionic and compact real Okubo products. The three algebra structures have the same additive lattice, the same positive composition norm and hence the same $240$ norm-one vectors, namely the roots of $E_8$ and the vertices of the Gosset polytope $4_{21}$. Their multiplicative structures are nevertheless different and this difference is already visible in the idempotents: among the same $240$ roots one finds respectively $1$, $57$ and $12$ nonzero integral idempotents. We show that these three counts organize the common root system in three increasingly polarized ways. The octonionic product leaves the full $E_8$ geometry unseparated; the $57$ para-octonionic idempotents reproduce the $E_7$ contact decomposition $1+56+126+56+1$; the $12$ Okubo idempotents split into four mutually orthogonal oriented $A_2$ triangles and hence determine canonically an $A_2^4$ subsystem. Choosing one triangle as the external $A_2$ then yields the $A_2+E_6$ Magic Star, while the remaining nine idempotents determine the trinification subsystem $A_2^3\\subset E_6$. This provides an arithmetic interpretation of the $E_8\\supset E_7$ and $E_8\\supset E_6+A_2$ decompositions directly from integral idempotents.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27762",
  "author_total": 2,
  "authors": [
    {
      "name": "Daniele Corradetti"
    },
    {
      "name": "Alessio Marrani"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/203pwkedghhrk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27762",
  "fulltext_url": null,
  "id": "203pwkedghhrk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27762",
  "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": "W7204799572",
  "pdf_url": "https://arxiv.org/pdf/2608.27762",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/203pwkedghhrk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.27762",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27762"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Integral Orders for the Okubo Algebra and Idempotent Geometries of the $E_8$ Lattice",
  "title_text": "Integral Orders for the Okubo Algebra and Idempotent Geometries of the E_8 Lattice",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/203pwkedghhrk",
  "venue": "arXiv",
  "versions": []
}