Generating pairs in ${\mathfrak{g}}_2$ related by an involution

Abstract

Let $\mathbb{O}$ be the real octonion division algebra, let ${\mathfrak{g}}_2=\mathrm{Der}(\mathbb{O})$, and fix a quaternionic subalgebra $H\subset \mathbb{O}$. We ask when an element $Z$ preserving $H$ and its conjugate $TZ{T}^{-1}$ generate ${\mathfrak{g}}_2$, where $T$ is a prescribed involutive octonion automorphism. We prove that such a $Z$ exists if and only if $T(H)\ne H$. Moreover, a fixed set of six derivations preserving $H$ contains a generating choice for every such $T$. For an arbitrary $Z\in {\mathfrak{g}}_2$, 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 $\mathfrak{su}(2)$. 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.

arXiv comment36 pages, 4 tables. Substantially revised and reorganized version; title changed. The exposition is self-contained relative to the technical companion. The revision adds prerequisite material, guided examples, explicit derivations of the two Gram certificates, and corrections and clarifications following referee review