On products of isometries of hyperbolic space

We show that for arbitrary fixed conjugacy classes C1,…,Cl, l⩾3, of loxodromic isometries of the two-dimensional complex or quaternionic hyperbolic space there exist isometries g1,…,gl, where each gi∈Ci, and whose product is the identity. The result follows from the properness, up to conjugation, of the multiplication map on a pair of conjugacy classes in rank 1 groups.