{
  "abstract": "We summarize a coordinate-free classification of six-dimensional real Lie algebras up to isomorphism. Instead of solving the overdetermined Jacobi system of 120 quadratic equations on 90 structure constants modulo $\\mathrm{GL}(6,\\mathbb{R})$, the method uses the Levi--Malcev decomposition, Mostow's nilradical constraints, and the Skjelbred--Sund cohomological central extension technique. In the semisimple sector, the four real forms of $A_1\\oplus A_1$ are classified by Killing signature invariants, with explicit matrix representatives for $\\mathfrak{sl}(2,\\mathbb{C})_{\\mathbb{R}}\\cong\\mathfrak{so}(3,1)$. For non-solvable algebras with three-dimensional radical, representation-theoretic rigidity shows that $\\mathfrak{su}(2)$ cannot couple indecomposably with non-abelian radicals, yielding exactly three indecomposable classes governed by $\\mathfrak{so}(3)$ and $\\mathfrak{sl}(2,\\mathbb{R})$. In the solvable category, Mostow's theorem restricts the nilradical dimension to $\\{3,4,5,6\\}$. The nilpotent sector in dimension six has twenty-four indecomposable isomorphism classes plus decomposable counterparts. Solvable non-nilpotent algebras are classified by nilradical codimension, with explicit real Jordan matrix representations for codimension one and two that resolve continuous moduli under gauge actions. The work also computes the minimal faithful representation dimension $μ(\\mathfrak{g})$ under Ado's theorem, relates the classification to invariant complex structures on nilmanifolds, and provides a deterministic, page-breakable identification algorithm.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.10009",
  "author_total": 1,
  "authors": [
    {
      "name": "Mehdi Nadjafikhah"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d2v339hv3wcat.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.10009",
  "fulltext_url": null,
  "id": "d2v339hv3wcat",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.10009",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.10009",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d2v339hv3wcat.ris",
  "sources": [
    {
      "first_seen": "2026-11-10T06:21:33Z",
      "native_id": "2611.10009",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.10009"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Classification of real Lie algebras of dimension six",
  "title_text": "Classification of real Lie algebras of dimension six",
  "type": "preprint",
  "updated_at": "2026-11-10T06:21:33Z",
  "url": "https://research.draconitic.com/paper/d2v339hv3wcat",
  "venue": "arXiv",
  "versions": []
}