 Solution

Use the standard ordered basis $({\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3)$ for both maps. The reflection in a hyperplane interchanges the second and third coordinates, so its matrix isOrient the rotation axis along $\mathbf{n}=(1,1,1)/\sqrt{3}$ and use the right-hand convention. The rotation matrix for angle $2\pi /3$ sends ${\mathbf{e}}_1$ to ${\mathbf{e}}_2$, ${\mathbf{e}}_2$ to ${\mathbf{e}}_3$, and ${\mathbf{e}}_3$ to ${\mathbf{e}}_1$. For example, the Rodrigues rotation formula givesMultiplying the rotation matrix by the dilation factor givesDirect multiplication gives $A^2=I$, $B^2=4\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right)$ and $B^3=8I$. Hence $B^3=8A^2$. If the opposite axis orientation is chosen, $B=2R^T$ also has cube $8I$. Independently of coordinates, two applications of the reflection give the identity, while three applications of the rotation-dilation give a full turn and dilation by $2^3$. Every basis represents these resulting maps by $I$ and $8I$. Thus the equality even holds when the two matrices are expressed in different bases.