## Orbits of action of the split group of type $F_4$

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.

Thank you.