Let $G$ be a group with a $\mathrm{BN}$-pair, and let $W:=N/(B\cap N)$ be the Weyl group of $G$, where $W$ is generated by a set $S$ of simple roots (as in the definition of a $\mathrm{BN}$-pair) and $(W,S)$ is a coxeter system with relations $R$. Suppose I have another generating set $S'$ for $W$ which satisfies the relations $R$, then how can I justify that the elements of $S'$ are also simple roots?
I know that coxeter groups with the same coxeter matrix are isomorphic so the groups generated by $S$ and $S'$ will be isomorphic. Is it true that every such isomorphism will send simple roots to simple roots? If yes, then why?
There often exists a Coxeter generating set that does not consist of simple reflections. For example, map $S\mapsto wSw^{-1}$ for some $w$. Usually there is a $w$ such that some element of $wSw^{-1}$ is not simple, but $(W,wSw^{-1})$ is an isomorphic Coxeter system. This answers your question in the negative.