Real analytic projective planes with large automorphism groups

We prove that there exist non-classical projective planes whose point space and line space are real analytic (or Nash) manifolds such that the geometric operations of joining points and intersecting lines are real analytic (even Nash) maps on their respective domains. Our examples have the dimensions 2, 4, or 8. These planes are the first examples of non-classical smooth projective planes with large automorphism groups. In dimension 2, they correspond to a class of projective planes discovered by Segre.