least 2, every lattice in G is an arithmetic group. The Galois action on the Dynkin diagram In seeking to classify reductive groups which need not be split, one step is the Tits index, which reduces the problem to the case of anisotropic groups. This reduction generalizes several fundamental theorems in algebra. For example, Witt's decomposition theorem says that a nondegenerate quadratic form over a field is determined up to isomorphism by its Witt index together with its anisotropic kernel. Likewise, the Artin–Wedderburn theorem reduces the classification of central simple algebras over a field to the case of division algebras. Generalizing these results, Tits showed that a reductive group over a field k is determined up to isomorphism by its Tits index together with its anisotropic kernel, an associated anisotropic semisimple k-group. For a reductive group G over a field k, the absolute Galois group Gal(ks/k) acts (continuously) on the "absolute" Dynkin diagram of G, that is, the Dynkin diagram of G over a separable closure ks (which is also the Dynkin diagram of G over an algebraic closure ). The Tits index of G consists of the root datum of Gks, the Galois action on its Dynkin diagram, and a Galois-invariant subset of the vertices of the Dynkin diagram. Traditionally, the Tits index is drawn by circling the Galois orbits in the given subset. There is a full classification of quasi-split groups in these terms. Namely, for each action of the absolute Galois group of a field k on a Dynkin diagram, there is a unique simply connected semisimple quasi-split group H over k with the given action. (For a quasi-split group, every Galois orbit in the Dynkin diagram is circled.) Moreover, any other simply connected semisimple group G over k with the given action is an inner form of the quasi-split group H, meaning that G is the group associated to an element of the Galois cohomology set H1(k,H/Z), where Z is the center of H. In other words, G is the twist of H associated to some H/Z-torsor over k, as discussed in the next section. Example: Let q be a nondegenerate quadratic form of even dimension 2n over a field k of characteristic not 2, with n ≥ 5. (These restrictions can be avoided.) Let G be the simple group SO(q) over k. The absolute Dynkin diagram of G is of type Dn, and so its