Reflection Groups on the Octave Hyperbolic Plane

For two different integral formsKof the exceptional Jordan algebra we show that Aut Kis generated by octave reflections. These provide “geometric” examples of discrete reflection groups acting with finite covolume on the octave (or Cayley) hyperbolic plane H2, the exceptional rank one symmetric space. (The isometry group of the plane is the exceptional Lie groupF4(−20).) Our groups are defined in terms of Coxeterʹs discrete subring of the nonassociative division algebra and we interpret them as the symmetry groups of “Lorentzian lattices” over . We also show that the reflection group of the “hyperbolic cell” over is the rotation subgroup of a particularrealreflection group acting onH8 H1. Part of our approach is the treatment of the Jordan algebra of matrices that are Hermitian with respect to any real symmetric matrix.