Let $k$ be a real closed field.
Let $G$ be a anisotropic algebraic group of type $F\_4$ over $k$.
Consider the projective, homogenous $F\_4$-variety $X\_4$ (Bourbaki enumeration).


To avoid confusion: Note that the possible Tits indexes of any $F\_4$ are anisotropic, split and $X\_4$ circled (which means the semisimple anisotropic kernel of $G$ is of type $B\_3$).


Question: Is $G$ split over the function field $k(X\_4)$ ?



When does F4 split over its homogeneous variety X_4?
A complete answer for questions of this kind is in my paper with Nikita Semenov "Generically split projective homogeneous varieties" in Duke (or refined version in J. K-Theory). In your situation this is the case if and only if G splits by a cubic field extension.


