{
  "abstract": "For a complex simple Lie algebra $\\mathfrak{g}$ with compact conjugation $τ$, Killing form $B$, and Hermitian form $H(x, y) = -B(x, τy)$, it is shown that $H([x,y],[x,y]) \\leq (h^{\\vee})^{-1}H(x,x)H(y,y)$, where $h^{\\vee}$ is the dual Coxeter number, with equality exactly for $H$-orthogonal pairs in $τ$-stable long-root subalgebras isomorphic to $\\mathfrak{sl}(2, \\mathbb{C})$. Equivalently, the comass of the Cartan $3$-form with respect to $H$ equals its comass on the compact real form: the spectral norm of the Cartan $3$-form does not increase under complexification. For the Frobenius norm on the image of a representation of Dynkin index $\\ell$ the optimal constant is $2/\\ell$; this contains the Böttcher--Wenzel and Bloch--Iserles inequalities, extends the latter to complex skew-symmetric matrices, and improves them substantially for the exceptional Lie algebras of types $F_{4}$, $E_{6}$, $E_{7}$, and $E_{8}$. The proof uses the Nahm algebra of $\\mathfrak{g}$ with the conjugation induced by $τ$: critical points of the relevant function correspond to $τ$-twisted idempotents, and its Hessian is governed by $τ$-twisted multiplication operators, whose anticommutation with multiplication by $i$ turns the one-sided second order condition at a maximum into a two-sided spectral bound. Subalgebras isomorphic to $\\mathfrak{sl}(2, \\mathbb{C})$ of Dynkin index $j$ give critical points with critical value $1/(jh^{\\vee})$, whose Morse indices are computed, and the set of maximum points is a nondegenerate critical manifold. Geometrically, the maximal complex sectional curvature of a compact simple Lie group equals its maximal sectional curvature.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11800",
  "author_total": 1,
  "authors": [
    {
      "name": "Daniel J. F. Fox"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8fv02w49yhkga.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11800",
  "fulltext_url": null,
  "id": "8fv02w49yhkga",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11800",
  "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.RA"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7221111477",
  "pdf_url": "https://arxiv.org/pdf/2610.11800",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8fv02w49yhkga.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.11800",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11800"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Sharp commutator bounds on complex simple Lie algebras: the Böttcher--Wenzel inequality and the comass of the Cartan $3$-form",
  "title_text": "Sharp commutator bounds on complex simple Lie algebras: the Böttcher--Wenzel inequality and the comass of the Cartan $3$-form",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/8fv02w49yhkga",
  "venue": "arXiv",
  "versions": []
}