Grassmann manifolds of Jordan algebras

We show that, in a JB-algebra, the projections form a Banach manifold and also, the rank-n projections in a JBW-factor form a Riemannian symmetric space of compact type, for $$n\, \in \,\user2{\mathbb{N}}\, \cup \, \left\{ 0 \right\}$$ .