More generally, the Tits index or Satake–Tits diagram of a reductive algebraic group over a field is a generalization of the Satake diagram to arbitrary fields, introduced by Jacques Tits (1966), that reduces the classification of reductive algebraic groups to that of anisotropic reductive algebraic groups.
Suppose that G is an algebraic group defined over a field k, such as the reals.
We let S be a maximal split torus in G, and take T to be a maximal torus containing S defined over the separable algebraic closure K of k. Then G(K) has a Dynkin diagram with respect to some choice of positive roots of T. This Dynkin diagram has a natural action of the Galois group of K/k.
Also some of the simple roots vanish on S. The Satake–Tits diagram is given by the Dynkin diagram D, together with the action of the Galois group, with the simple roots vanishing on S colored black.
Both Satake and Vogan diagrams are used to classify semisimple Lie groups or algebras (or algebraic groups) over the reals and both consist of Dynkin diagrams enriched by blackening a subset of the nodes and connecting some pairs of vertices by arrows.
Satake diagrams, however, can be generalized to any field (see above) and fall under the general paradigm of Galois cohomology, whereas Vogan diagrams are defined specifically over the reals.
Generally speaking, the structure of a real semisimple Lie algebra is encoded in a more transparent way in its Satake diagram, but Vogan diagrams are simpler to classify.
The essential difference is that the Satake diagram of a real semisimple Lie algebra
(the +1 and −1 eigenspaces of θ) is defined by starting from a maximally noncompact θ-stable Cartan subalgebra
appears as the Lie algebra of the maximal split torus S), whereas Vogan diagrams are defined starting from a maximally compact θ-stable Cartan subalgebra, that is, one for which