I’m just beginning to learn about Coxeter groups and at one point the textbook’s author seems to take it for grant that each generator cannot be “reduced” into a product of other generators. Specifically, it seems the following claim is so obvious that it need not be stated:
If $\{s_1, s_2,..., s_n\}$ is the set of generators of a coxeter group $W$, then one cannot have $s_i=s_1s_2...s_{j-1}s_js_{j-1}...s_1$ for some $j$ without $i, j$ being $1$.
So I suspected something like $s_1=s_2s_4s_8$ cannot happen too. I know the question looks really stupid but it seems I’m missing something fundamental.