Special prehomogeneous vector spaces associated to F4, E6, E7, E8 and simple Jordan algebras of rank 3

A simple complex Lie algebra has a natural grading associated to the highest root. It is related to the quaternionic real form of . The real rank of the associated symmetric space is shown to be at most 4. When the rank is equal to 4 (i.e., ), the semi-simple Lie algebra is shown to be the conformal algebra of a rank 3 semi-simple Jordan algebra . If is one of the four exceptional Lie algebras F4,E6,E7,E8, then is simple. The vector space is a prehomogeneous vector space under the action of , where is the adjoint group of . It admits a non-zero M-invariant polynomial of degree 4. Conversely, to any simple Jordan algebra of rank 3 is naturally associated a representation of (a twofold covering of) its conformal group on a vector space , such that is a prehomogeneous vector space under the action of . An invariant polynomial of degree 4 is explicitly constructed. A geometric description of all the orbits of in is given.