symmetric spaces with G = H × H. Any semisimple symmetric space is a product of symmetric spaces of this form with symmetric spaces such that is simple. It remains to describe the latter case. For this, one needs to classify involutions σ of a (real) simple Lie algebra . If is not simple, then is a complex simple Lie algebra, and the corresponding symmetric spaces have the form G/H, where H is a real form of G: these are the analogues of the Riemannian symmetric spaces G/K with G a complex simple Lie group, and K a maximal compact subgroup. Thus we may assume is simple. The real subalgebra may be viewed as the fixed point set of a complex antilinear involution τ of , while σ extends to a complex antilinear involution of commuting with τ and hence also a complex linear involution σ∘τ. The classification therefore reduces to the classification of commuting pairs of antilinear involutions of a complex Lie algebra. The composite σ∘τ determines a complex symmetric space, while τ determines a real form. From this it is easy to construct tables of symmetric spaces for any given , and furthermore, there is an obvious duality given by exchanging σ and τ. This extends the compact/non-compact duality from the Riemannian case, where either σ or τ is a Cartan involution, i.e., its fixed point set is a maximal compact subalgebra. Tables The following table indexes the real symmetric spaces by complex symmetric spaces and real forms, for each classical and exceptional complex simple Lie group. For exceptional simple Lie groups, the Riemannian case is included explicitly below, by allowing σ to be the identity involution (indicated by a dash). In the above tables this is implicitly covered by the case kl=0. Weakly symmetric Riemannian spaces In the 1950s Atle Selberg extended Cartan's definition of symmetric space to that of weakly symmetric Riemannian space, or in current terminology weakly symmetric space. These are defined as Riemannian manifolds M with a transitive connected Lie group of isometries G and an isometry σ normalising G such that given x, y in M there is an isometry s in G such that sx = σy and sy = σx. (Selberg's assumption that σ2 should be an element of G was later shown to be unnecessary by Ernest Vinberg.) Selberg proved that weakly symmetric spaces give rise to Gelfand pairs, so