I have an infinite reflection group https://en.wikipedia.org/wiki/Coxeter_group
Take for example the affine groups $[4,4],[4,3,4],[4,3,3,4]$...
I'd like to get an explicit expression for generators that generate the translation subgroup: $$T=\langle t_1,\cdots,t_r \; :\; t_i t_j = t_j t_i\rangle $$ and each $t_i$ has infinite order.
For example for [4,4] : $t_1=r_1r_2r_3; t_2=r_3r_2r_1$;
[Edit : per Derek Holt's comment $t_1=r_2r_3r_2r_1; t_2=r_3r_2r_1r_2;$ is the appropriate example]
What about others in general? I'm sure this is a known result but I can't find it.