  Similar sublattices of the root lattice $A_4$ are possible, according to a
result of Conway, Rains and Sloane, for each index that is the square of a
non-zero integer of the form $m^2 + mn - n^2$. Here, we add a constructive
approach, based on the arithmetic of the quaternion algebra $\mathbb{H}
(\mathbb{Q} (\sqrt{5}))$ 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.
