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?