text: The subgroup structure suggests another way to compose an arbitrary isometry:
  : Pick a fixed point, and a mirror through it.
  # If the isometry is odd, use the mirror; otherwise do not.
  # If necessary, rotate around the fixed point.
  # If necessary, translate.
  This works because translations are a normal subgroup of the full group of isometries, with quotient the orthogonal group; and rotations about a fixed point are a normal subgroup of the orthogonal group, with quotient a single reflection.