The Magic Square of Reflections And Rotations

The title refers to the square diagram of bijections between various classes of finite subgroups of classical groups. We survey how Coxeter's work implies a bijection between complex reflection groups of rank two and real reflection groups in O(3). We also consider this magic square of reflections and rotations in the framework of Clifford algebras: we give an interpretation using (s)pin groups and explore these groups in small dimensions.