Kirillov and Berenstein, in their article "Groups generated by involutions, Gelfand-Tsetlin patterns and combinatorics of Young tableaux" (available at math.uoregon.edu/~arkadiy/bk1.pdf), present a construction that, starting from involutions $t_1,t_2,\dots,t_n$ satisfying $(t_i t_j)^2 = 1$ for $|i-j|>1$ and $(t_i t_j)^6 = 1$ for $|i-j|=1$, create new involutions $s_1,s_2,\dots$ satisfying $(s_i s_j)^2 = 1$ for $|i-j|>1$ and $(s_i s_j)^3 = 1$ for $|i-j|=1$; specifically, they define $s_i = q_i t_1 q_i^{-1}$ with $q_i = (t_1)(t_2 t_1) \cdots (t_i t_{i-1} \cdots t_2 t_1)$.

Unfortunately K&B do not prove directly that their construction has the desired properties, but rather derive it from properties of a more complicated construction. (See page 18: "Part 1^{0} of Theorem 1.1 follows from part 2^{0} and the equality (1.18)." Theorem 1.1 itself appears on page 16.)

Can anyone give a self-contained explanation of why this trick works?

(Is it a special case of a more general trick that, given a Coxeter group, constructs a subgroup whose diagram is similar to that of the original, but simpler beause some of the Coxeter exponents are replaced by proper divisors of those exponents?)

I have verified the claim up through $(s_2 s_3)^3 = 1$, but even proving $(s_3 s_4)^3 = 1$ is giving me trouble.