Spin(9) stabilizer of an octonionic spin factor

Statement

Let $J=H_3(\mathbb O)$ , let $F_4=\operatorname{Aut}(J)$ , and let $A\subset J$ be a Jordan subalgebra isomorphic to $H_2(\mathbb O)$ . Then its setwise stabilizer is connected and

\operatorname{Stab}_{F_4}(A)\cong\mathrm{Spin}(9).

Reduction to a primitive idempotent

There is a unique trace-two idempotent $\ell$ with $A=J_1(\ell)$ . Its complement $p=1-\ell$ is a primitive idempotent, and an automorphism preserves $A$ if and only if it preserves $p$ . Thus this stabilizer is the isotropy group of a point of the octonionic projective plane, and the transitive orbit description is

\mathbb O P^2\cong F_4/\mathrm{Spin}(9).

Here $F_4$ and $\mathrm{Spin}(9)$ denote their compact real forms.

The connectedness is unlike the stabilizer of a complex-qutrit subalgebra, whose full stabilizer has more than one connected component.

References

Ichirô Yokota, Exceptional Lie Groups, 2009, Chapter 2. arXiv:0902.0431.
Tonny A. Springer and Ferdinand D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups, Springer, 2000, Chapters 7 and 8. Publisher record.
John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026, §§3 and 5. arXiv:2606.15235.