Re ection Groups on the Octave Hyperbolic

For two diierent integral forms K of the exceptional Jordan algebra we show that Aut K is generated by octave reeections. These providègeometric' examples of discrete reeection groups acting with nite covolume on the octave (or Cayley) hyperbolic plane O H 2 , the exceptional rank one symmetric space. (The isometry group of the plane is the exceptional Lie group F 4(?20) .) Our groups are deened in terms of Coxeter's discrete subring K of the nonassociative division algebra O and we interpret them as the symmetry groups of \Lorentzian lattices" over K. We also show that the reeection group of the \hyperbolic cell" over K is the rotation subgroup of a particular real reeection group acting on H 8 = O H 1. Part of our approach is the treatment of the Jordan algebra of matrices that are Hermitian with respect to any real symmetric matrix.