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 $1\leq i\leq n-1$
• $m_{n(n+1)}=4$
• $m_{ij}=2$ 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 $1$.