Among the sublattices of a given lattice in Euclidean space, those similar to the original lattice form an interesting and important subclass. In recent years, several attempts have been made to classify them, with limited success so far. An exception are lattices in dimensions up to 4, and some of the root lattices. The latter were studied by Conway, Rains and Sloane in a non- constructive manner by means of quadratic forms, which gave access to the possible sublattice indices, but not to the sublattices themselves. In particular, the root lattice $A_4$ could not be treated completely. It is the purpose of this talk to add a constructive approach, based on the arithmetic of a certain quaternion algebra and the existence of an unusual involution of the second kind. This also provides the actual sublattices and the number of different solutions for a given index. The corresponding Dirichlet series generating function is closely related to the zeta function of the icosian ring. This is joint work with Manuela Heuer and Robert V. Moody. Michael Baake (Bielfeld) Note: the JC organizers apologize for the short notice of this talk.