Higher rank curved Lie triples

A substantial proper submanifold M of a Riemannian symmetric space S is called a curved Lie triple if its tangent space at every point is invariant under the curvature tensor of S, i.e. a sub-Lie triple. E.g. any complex submanifold of complex projective space has this property. However, if the tangent Lie triple is irreducible and of higher rank, we show a certain rigidity using the holonomy theorem of Berger and Simons: M must be intrinsically locally symmetric. In fact we conjecture that M is an extrinsically symmetric isotropy orbit. We are able to prove this conjecture provided that a tangent space of M is also a tangent space of such an orbit.