Group of Reflection Matrices Order 4

Definition
Consider the algebraic structure $S$ of reflection matrices:
 * $R_4 = \set {\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}, \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} }$

under the operation of (conventional) matrix multiplication.

$R_4$ is the group of reflection matrices of order $4$.

Also see

 * Group of Reflection Matrices Order 4 is Klein Four-Group