DOUBLE VOGAN DIAGRAMS AND SEMISIMPLE SYMMETRIC SPACES

A Vogan diagram is a set of involution and painting oil a Dynkin diagram. It selects a real form, or equivalently all involution, from a complex simple Lie algebra. We introduce the double Vogan diagram, which is two sets of Vogan diagrams superimposed on an affine Dynkin diagram. They correspond to pairs of commuting involutions on complex simple Lie algebras, and therefore provide an independent classification of the simple locally symmetric pairs.