Herglotz concluded, that any such substitution corresponds to a Lorentz transformation, establishing a one-to-one correspondence to hyperbolic motions in R. The relation between the Lorentz group and the Cayley metric in hyperbolic space was also pointed out by Klein (1910) as well as Pauli (1921). The corresponding isomorphism of the Möbius group to the Lorentz group was employed, among others, by Roger Penrose.