Similar sublattices of the root lattice A4

Similar sublattices of the root lattice A4 are possible [J.H. Conway, E.M. Rains, N.J.A. Sloane, On the existence of similar sublattices, Can. J. Math. 51 (1999) 1300–1306] for each index that is the square of a non-zero integer of the form m2+mn−n2. Here, we add a constructive approach, based on the arithmetic of the quaternion algebra and the existence of a particular involution of the second kind, which 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.