if it contains a nontrivial split torus), and otherwise anisotropic. For a semisimple group G over a field k, the following conditions are equivalent: G is isotropic (that is, G contains a copy of the multiplicative group Gm over k); G contains a parabolic subgroup over k not equal to G; G contains a copy of the additive group Ga over k. For k perfect, it is also equivalent to say that G(k) contains a unipotent element other than 1. For a connected linear algebraic group G over a local field k of characteristic zero (such as the real numbers), the group G(k) is compact in the classical topology (based on the topology of k) if and only if G is reductive and anisotropic. Example: the orthogonal group SO(p,q) over R has real rank min(p,q), and so it is anisotropic if and only if p or q is zero. A reductive group G over a field k is called quasi-split if it contains a Borel subgroup over k. A split reductive group is quasi-split. If G is quasi-split over k, then any two Borel subgroups of G are conjugate by some element of G(k). Example: the orthogonal group SO(p,q) over R is split if and only if |p−q| ≤ 1, and it is quasi-split if and only if |p−q| ≤ 2. Structure of semisimple groups as abstract groups For a simply connected split semisimple group G over a field k, Robert Steinberg gave an explicit presentation of the abstract group G(k). It is generated by copies of the additive group of k indexed by the roots of G (the root subgroups), with relations determined by the Dynkin diagram of G. For a simply connected split semisimple group G over a perfect field k, Steinberg also determined the automorphism group of the abstract group G(k). Every automorphism is the product of an inner automorphism, a diagonal automorphism (meaning conjugation by a suitable -point of a maximal torus), a graph automorphism (corresponding to an automorphism of the Dynkin diagram), and a field automorphism (coming from an automorphism of the field k). For a k-simple algebraic group G, Tits's simplicity theorem says that the abstract group G(k) is close to being simple, under mild assumptions. Namely, suppose that G is isotropic over k, and suppose that the field k has at least 4 elements. Let G(k)+ be the subgroup of the abstract group