Geometries over the algebra of antioctaves

We give a rigorous construction for a projective and a noneuclidean geometry over the alternative algebra of antioctaves (split octaves). This construction generalizes Freudenthal's definition of the projective plane over the algebra of octaves (Cayley numbers). It is proved that the groups of automorphisms of the projective and the noneuclidean plane are simple noncompact Lie groups of types $E_6$ and $F_4$, respectively.