Paul Schwahn on Nostr: I didn't know about the transitive action of F₄ on Jordan frames, but that seems ...

I didn't know about the transitive action of F₄ on Jordan frames, but that seems very useful! I also think the subgroup fixing each of e₁, e₂, e₃ is Spin(8). Here's a sketch of the argument:

The stabilizer of e₁ is Spin(9), and this also preserves the 𝔥₂(𝕆)-subalgebra containing e₂+e₃. I think you have shown somewhere else that this Spin(9) acts on this 𝔥₂(𝕆) by automorphisms, via the double covering Spin(9) → SO(9) = Aut 𝔥₂(𝕆).

Now we are one level lower: the set of points (trace 1 idempotents) in 𝔥₂(𝕆) is 𝕆ℙ¹ ≅ SO(9)/SO(8) ≅ Spin(9)/Spin(8), so the subgroup of Spin(9) fixing in addition the point e₂ (and therefore its complement e₃) must be Spin(8).

There is another thing I would like to understand: whether there is an incidence-geometric way of seeing what (SU(3) × SU(3))/ℤ₃ stabilizes. As before, Spin(9) is the stabilizer in F₄ of a trace 1 idempotent, so the set of points of 𝕆ℙ² can be identified with F₄/Spin(9).

We mused earlier that F₄/(SU(3) × SU(3))/ℤ₃ may be the set of "ℂℙ²s in 𝕆ℙ²" - but what even is a ℂℙ² in 𝕆ℙ²? I want to have a definition like this:

"An 𝕆ℙ¹⊂𝕆ℙ² is the set of all trace 1 idempotents in 𝔥₃(𝕆) incident with a given trace 2 idempotent."
"A ℂℙ²⊂𝕆ℙ² is the set of all trace 1 idempotents in 𝔥₃(𝕆) satisfying [condition]".