F4 transitivity on compatible Jordan-subalgebra pairs

Statement

Let $J=H_3(\mathbb O)$ . The compact group $F_4=\operatorname{Aut}(J)$ acts transitively on ordered pairs $(A,B)$ of Jordan subalgebras satisfying

A\cong H_2(\mathbb O), \qquad B\cong H_3(\mathbb C), \qquad A\cap B\cong H_2(\mathbb C).

All three hypotheses, including the intersection condition, are part of the statement.

Equivalently, $F_4$ acts transitively on incident pairs $(X,B)$ with $X\subset B\subset J$ , $X\cong H_2(\mathbb C)$ , and $B\cong H_3(\mathbb C)$ : the passage between the two forms is

X=A\cap B, \qquad A=J_1(1_X),

where the second formula is the unique octonionic spin-factor corner containing $X$ .

Proof outline

Use transitivity on complex-qutrit subalgebras to make the two $B$ 's equal to the standard $H_3(\mathbb C)$ . Within this subalgebra, $\mathrm{SU}(3)$ acts transitively on trace-two idempotents, equivalently on complex two-planes in $\mathbb C^3$ , and hence on its $H_2(\mathbb C)$ -corners. This action extends through $\operatorname{Stab}_{F_4}(B)^0$ . Uniqueness of the associated $H_2(\mathbb O)$ -corner then carries $A$ to $A'$ .

Consequence for stabilizer intersections

The transitivity says that a stabilizer computation for one standard pair applies to every compatible pair. In particular it is the geometric step behind the characterization of the Standard Model gauge group as an $F_4$ stabilizer intersection, where the identity component of the qutrit stabilizer must be used.

References

John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026, Lemma 11 and Theorems 1–2. arXiv:2606.15235.
Ichirô Yokota, Exceptional Lie Groups, 2009, Chapter 2. arXiv:0902.0431.