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PreprintCC BY 4.0arXiv

Generating pairs in g2 related by an involution

  • Santiago Pineda Montoya
  • Johan H. Rúa Muñoz
  • Borut Jurčič Zlobec
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Abstract

Let 𝕆 be the real octonion division algebra, let g2=Der(𝕆), and fix a quaternionic subalgebra H⊂𝕆. We ask when an element Z preserving H and its conjugate TZT−1 generate g2, where T is a prescribed involutive octonion automorphism. We prove that such a Z exists if and only if T(H)≠ H. Moreover, a fixed set of six derivations preserving H contains a generating choice for every such T. For an arbitrary Z∈g2, we construct a polynomial criterion with two factors: one detects reducibility on the imaginary octonions, and the other detects generated algebras of dimension at most three. An irreducible proper generated algebra must be a principal su(2). We characterize the elements admitting this case by their rotation frequencies and parametrize all corresponding involutions. For one specified involution and quaternionic subalgebra H, we describe the nongenerating set by four geometric families and six irredundant systems of scalar equations, and prove that it has real dimension four.

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Cite

BibTeXRIS
BibTeX
@misc{montoya2026generating,
  title = {{Generating pairs in $\mathfrak g_2$ related by an involution}},
  author = {Santiago Pineda Montoya and Johan H. Rúa Muñoz and Borut Jurčič Zlobec},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.15497},
  eprint = {2609.15497},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.15497},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Generating pairs in g_2 related by an involution
AU  - Santiago Pineda Montoya
AU  - Johan H. Rúa Muñoz
AU  - Borut Jurčič Zlobec
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.15497
UR  - https://arxiv.org/abs/2609.15497
AB  - Let 𝕆 be the real octonion division algebra, let g_2=Der(𝕆), and fix a quaternionic subalgebra H⊂𝕆. We ask when an element Z preserving H and its conjugate TZT^−1 generate g_2, where T is a prescribed involutive octonion automorphism. We prove that such a Z exists if and only if T(H)≠ H. Moreover, a fixed set of six derivations preserving H contains a generating choice for every such T. For an arbitrary Z∈g_2, we construct a polynomial criterion with two factors: one detects reducibility on the imaginary octonions, and the other detects generated algebras of dimension at most three. An irreducible proper generated algebra must be a principal su(2). We characterize the elements admitting this case by their rotation frequencies and parametrize all corresponding involutions. For one specified involution and quaternionic subalgebra H, we describe the nongenerating set by four geometric families and six irredundant systems of scalar equations, and prove that it has real dimension four.
ER  -