connected group and A is a k-subgroup scheme of the center of G. For example, the simply connected split simple groups over a field k corresponding to the "classical" Dynkin diagrams are as follows: An: SL(n+1) over k; Bn: the spin group Spin(2n+1) associated to a quadratic form of dimension 2n+1 over k with Witt index n, for example the form Cn: the symplectic group Sp(2n) over k; Dn: the spin group Spin(2n) associated to a quadratic form of dimension 2n over k with Witt index n, which can be written as: The outer automorphism group of a split reductive group G over a field k is isomorphic to the automorphism group of the root datum of G. Moreover, the automorphism group of G splits as a semidirect product: where Z is the center of G. For a split semisimple simply connected group G over a field, the outer automorphism group of G has a simpler description: it is the automorphism group of the Dynkin diagram of G. Reductive group schemes A group scheme G over a scheme S is called reductive if the morphism G → S is smooth and affine, and every geometric fiber is reductive. (For a point p in S, the corresponding geometric fiber means the base change of G to an algebraic closure of the residue field of p.) Extending Chevalley's work, Michel Demazure and Grothendieck showed that split reductive group schemes over any nonempty scheme S are classified by root data. This statement includes the existence of Chevalley groups as group schemes over Z, and it says that every split reductive group over a scheme S is isomorphic to the base change of a Chevalley group from Z to S. Real reductive groups In the context of Lie groups rather than algebraic groups, a real reductive group is a Lie group G such that there is a linear algebraic group L over R whose identity component (in the Zariski topology) is reductive, and a homomorphism G → L(R) whose kernel is finite and whose image is open in L(R) (in the classical topology). It is also standard to assume that the image of the adjoint representation Ad(G) is contained in Int(gC) = Ad(L0(C)) (which is automatic for G connected). In particular, every connected semisimple Lie group (meaning that its Lie algebra is semisimple) is reductive. Also, the Lie group R is reductive in this