Definition 1

A Tits system is a quadruple $(G, B, N, S)$, where G is a group, B and N are two subgroups of G and S is a subset of $N/(B \cap N)$, satisfying the following axioms:

(T1) The set $B \cup N$ generates G and $B \cap N$ is a normal subgroup of N. (T2) The set S generates the group $W = N/(B \cap N)$ and consists of elements of order 2. (T3) $sBw \subset BwB \cup BswB$ for $s \in S$ and $w \in W$. (T4) For all $s \in S$, $sBs \notin B$.

The group $W = N/(B \cap N)$ is sometimes called the Weyl group of the Tits system $(G, B, N, S)$.

5 Every element of W is a coset modulo $B \cap N$, and is thus a subset of G; hence products such as $BwB$ make sense. More generally, for any subset A of W, we denote by BAB the subset $\bigcup_{w \in A} BwB$.