Quaternionic Reflections and Lie Generation in $G_2$

Abstract

Let $\mathbb{O}$ be the real octonion division algebra and ${\mathfrak{g}}_2=\mathrm{Der}(\mathbb{O})$. A quaternionic reflection $T\in G_2=\mathrm{Aut}(\mathbb{O})$ fixes a quaternionic subalgebra pointwise and negates its orthogonal complement. For the associated compact symmetric pair ${\mathfrak{g}}_2={\mathfrak{k}}_T\oplus {\mathfrak{m}}_T$, we give an explicit factored polynomial criterion for generation by an independently chosen even element and odd element. Equivalently, for the reflected pair $(Z,TZT)$ with $Z\in {\mathfrak{g}}_2$, the criterion is ${\mathcal{U}}_{38}={\mathcal{R}}_{30}{\mathcal{H}}_8>0$. The degree-30 Gram determinant detects reducibility on $\mathrm{Im}\mathbb{O}$, while the degree-8 factor detects generated algebras of dimension at most three. A principal $\mathfrak{su}(2)$ is the remaining irreducible proper possibility. Such a reflected principal closure is realizable for a nonzero $Z$ if and only if its positive rotation frequencies have ratio $1:2:3$; for each such $Z$, all realizing reflections are parametrized by a two-torus and an open interval. A different restriction arises when $Z$ must lie in the stabilizer algebra ${\mathfrak{k}}_A\simeq \mathfrak{so}(4)$ of a fixed quaternionic subalgebra $H_A$. For every reflection moving $H_A$, a fixed list of six elements of ${\mathfrak{k}}_A$ contains a generating choice; reflections preserving $H_A$ admit none. For one explicit relative configuration $({\mathfrak{k}}_A,S)$, we describe the entire nongenerating set by four geometric families and by six irredundant systems of scalar equations. We determine its real dimension and all generated algebras of dimension at most three; a homogeneous degree-34 polynomial packages the six tests. The degree is not asserted to be canonical or minimal. Two explicit local coordinate maps use short words in a finite-time pulse and one reflection, whose generated subgroup is dense in $G_2$.

53 pages, 1 figure, 2 tables