Let the split group of type $F\_4$ act as the automorphism group of the split Albert algebra $A$. Consider the action of $F\_4\times \mathbb{G}\_m$ on $A$, given by letting $\mathbb{G}\_m$ act by scalar multiplication. 


Does this action have finitely many orbits? Are the stabilizers known? I am very new to $F\_4$ and I am slowly going through the basics, but a reference in this direction would greatly simplify my life.


Edit: I now realize that the answer to my question is negative, because the rational quotient $A\_0/F\_4$ is $2$-dimensional (where $A\_0$ is the trace-zero subspace of $A$), just by looking at diagonal matrices. Possibly the answer below refers to $E\_6\times \mathbb G\_m$?


So the correct question is: are there finitely many orbits in the locus of matrices not diagonalizable with distinct eigenvalues? What are their stabilizers?



Orbits of action of the split group of type $F_4$
I work over the complexe numbers, but the story is similar over any fields (provided you choose the split octonions).
Let us identify the Albert algebra $A$ with the algebra of self-adjoint $3\*3$ matrices with octonionic coefficients.
The hyperplane given by $\mathrm{Tr}(X) = 0$ is stabilized by $F\_4 \times \mathbb{G}\_m$ and the action of $F\_4 \times \mathbb{G}\_m$ on this hyperplane can be devided into two types.


First : the orbits located in the discriminant hypersurface (that is the set of matrices of rank less or equal to $2$). They are:


\_the zero of $A$,


\_the set of matrice of rank $1$ (denote it by $Z\_0$, it has dimension $16$),


\_an orbit which closure contains the previous one and that has dimension $17$,


\_the (open part) of a certain *restricted* tangent variety to $Z\_0$,


\_the discriminant hypersurface minus the previous one and the closure of the third one : it has dimension $25$.


Note that the equation of the discriminant hypersurface is $6\mathrm{det}(X)^2-9(\mathrm{Tr}(X^2))^3 = 0$.


Now, let's turn to the orbits located outside of the discriminant hypersurface. There is a one dimensional family of them: they are the hypersurfaces given by the equation $6t\mathrm{det}(X)^2-s(\mathrm{Tr}(X^2))^3 = 0$, for $[s,t] \in \mathbb{P}^1 \backslash [1,9]$.


All of this is discussed at lentgh and proved in details in proposition 3.5 of <https://arxiv.org/pdf/math/0306328.pdf>


