We adress the problem of the reasons for the existence of 12 symmetric spaces with the exceptional Lie groups. The 1+2 cases for G2 and F4 respectively are easily explained from the octonionic nature of these groups. The 4+3+2 cases on the E6,7,8 series require the magic square of Freudenthal and, for the split case, an appeal to the supergravity chain in 5, 4 and 3 spacetime dimensions.

Let θ : g → g be an involution of a complex semisimple Lie algebra, k ⊂ g the fixed points of θ, and V = g/k the corresponding symmetric space. The adjoint form K of k naturally acts on V . The orbits and invariants of this representation were studied by Kostant and Rallis in [KR]. Let X = K\\V be the invariant theory quotient, and f : V → X be the quotient map. The space X is isomorphic to C. ...

Let r be finite connected and G a group of automorphisms of r which is transitive on vertices. Suppose that, for a vertex 0 of r, S ~ G~(O') ::; Aut S for some simple group S with S acting primitively on the set r( a) of neighbours of 0, and suppose that G is minimal with these properties. Then one of: (i) G is a nonabelian simple group, (ii) r is a Cayley graph for a normal subgroup N of G and...