This chapter introduces the notion of a Tits index and the notion of the relative Coxeter diagram of a Tits index. It first defines a Tits index, which can be anisotropic or isotropic, quasi-split or split, before considering a number of propositions regarding compatible representations. It then gives a proof of the theorem that includes two assumptions about a Coxeter system, focusing on the absolute Coxeter system, the relative Coxeter system, and the relative Coxeter group of the Tits index, as well as the absolute Coxeter diagram (or absolute type), the relative Coxeter diagram (or relative type), and the absolute rank and the relative rank of the Tits index. The chapter concludes with some observations about the case that (W, S) is spherical, irreducible or affine.