space V of dimension n: For the orthogonal group or the symplectic group, the projective homogeneous varieties have a similar description as varieties of isotropic flags with respect to a given quadratic form or symplectic form. For any reductive group G with a Borel subgroup B, G/B is called the flag variety or flag manifold of G. Classification of split reductive groups Chevalley showed in 1958 that the reductive groups over any algebraically closed field are classified up to isomorphism by root data. In particular, the semisimple groups over an algebraically closed field are classified up to central isogenies by their Dynkin diagram, and the simple groups correspond to the connected diagrams. Thus there are simple groups of types An, Bn, Cn, Dn, E6, E7, E8, F4, G2. This result is essentially identical to the classifications of compact Lie groups or complex semisimple Lie algebras, by Wilhelm Killing and Élie Cartan in the 1880s and 1890s. In particular, the dimensions, centers, and other properties of the simple algebraic groups can be read from the list of simple Lie groups. It is remarkable that the classification of reductive groups is independent of the characteristic. For comparison, there are many more simple Lie algebras in positive characteristic than in characteristic zero. The exceptional groups G of type G2 and E6 had been constructed earlier, at least in the form of the abstract group G(k), by L. E. Dickson. For example, the group G2 is the automorphism group of an octonion algebra over k. By contrast, the Chevalley groups of type F4, E7, E8 over a field of positive characteristic were completely new. More generally, the classification of split reductive groups is the same over any field. A semisimple group G over a field k is called simply connected if every central isogeny from a semisimple group to G is an isomorphism. (For G semisimple over the complex numbers, being simply connected in this sense is equivalent to G(C) being simply connected in the classical topology.) Chevalley's classification gives that, over any field k, there is a unique simply connected split semisimple group G with a given Dynkin diagram, with simple groups corresponding to the connected diagrams. At the other extreme, a semisimple group is of adjoint type if its center is trivial. The split semisimple groups over k with given Dynkin diagram are exactly the groups G/A, where G is the simply