F4 transitivity on complex-qutrit subalgebras

Statement

Let $J=H_3(\mathbb O)$ be the compact real Albert algebra. Its automorphism group $F_4$ acts transitively on the set

\{B\subset J:B\text{ is a Jordan subalgebra and }B\cong H_3(\mathbb C)\}.

Equivalently, every complex-qutrit Jordan subalgebra can be carried to the standard $H_3(\mathbb C)\subset H_3(\mathbb O)$ by an automorphism of $J$ .

Proof mechanism

First use transitivity of $F_4$ on Jordan frames to arrange that $B$ contains the standard frame. The associated three octonionic two-planes can then be normalized by the pointwise frame stabilizer $\mathrm{Spin}(8)$ . Its vector representation makes the first two-plane the standard $\mathbb C\subset\mathbb O$ ; the residual $\mathrm{Spin}(6)\cong\mathrm{SU}(4)$ acts transitively on the unit sphere in a half-spin representation and normalizes the remaining parameter. The resulting subalgebra is the standard one.

Orbit and stabilizer

For a fixed $B$ , the orbit is the homogeneous space

F_4/\operatorname{Stab}_{F_4}(B).

Its isotropy group is not connected. Only its identity component is $(\mathrm{SU}(3)\times\mathrm{SU}(3))/\mu_3$ , as described in the complex-qutrit stabilizer theorem.

References

John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026, Lemma 6. arXiv:2606.15235.
John F. Adams, Lectures on Exceptional Lie Groups, University of Chicago Press, 1996, Chapters 1–3. Publisher record.