Symmetric spaces as Grassmannians

In the PhD thesis of Huang with Leung (Huang, A uniform description of Riemannian symmetric spaces as Grassmannians using magic square, www.ims.cuhk.edu.hk/~leung/ ; Huang and Leung, Math Ann 350:76–106, 2010 ), all compact symmetric spaces are represented as (structured) Grassmannians over the algebra where are real division algebras. This was known in some (infinitesimal) sense for exceptional spaces (see Baez, Bull Am Math Soc 39:145–205, 2001 ); the main purpose in Huang ( www.ims.cuhk.edu.hk/~leung/ ) and Huang and Leung (Math Ann 350:76–106, 2010 ) was to give a similar description for the classical spaces. In the present paper we give a different approach to this result by investigating the fixed algebras of involutions on with half-dimensional eigenspaces together with the automorphism groups of and . We also relate the results to the classification of self-reflective submanifolds in Chen and Nagano (Trans Am Math Soc 308:273–297, 1988 ) and Leung (J Differ Geom 14:167–177, 1979 ).