We prove that a half dimensional, totally real and totally geodesic submanifold of a compact Riemannian 3-symmetric space is expressed as an orbit of a Lie subgroup of the isometry group of the ambient manifold. Moreover, we associate such submanifolds with graded Lie algebras of the second kind.
"Totally real totally geodesic submanifolds of compact 3-symmetric spaces." Tohoku Math. J. (2) 53 (1) 131 - 143, 2001. https://doi.org/10.2748/tmj/1178207535