Signed symmetric group of finite degree is a Coxeter group

Statement
Suppose $$n$$ is a nonnegative integer and $$G$$ is the signed symmetric group of degree $$n + 1$$. Then, $$G$$ is isomorphic to a Coxeter group with generators $$s_1,s_2,\dots,s_n,t = s_{n+1}$$, the $$m_{ij}$$ are as follows:


 * $$m_{i(i+1)} = 3$$ for $$1 \le i \le n - 1$$
 * $$m_{n(n+1)} = 4$$
 * $$m_{ij} = 2$$ if $$|i - j| > 1$$

Under the isomorphism, $$s_i$$ is idnetified with the transposition $$(i,i+1)$$ and $$t = s_{n+1}$$ is the diagonal matrix with the last entry $$-1$$ and all remaining entries equal to $$1$$.