  Recently, the group of coincidence isometries of the root lattice $A_4$ has
been determined providing a classification of these isometries with respect to
their coincidence indices. A more difficult task is the classification of all
CSLs, since different coincidence isometries may generate the same CSL. In
contrast to the typical examples in dimensions $d \leq 3$, where coincidence
isometries generating the same CSL can only differ by a symmetry operation, the
situation is more involved in 4 dimensions. Here, we find coincidence
isometries that are not related by a symmetry operation but nevertheless give
rise to the same CSL. We indicate how the classification of CSLs can be
obtained by making use of the icosian ring and provide explicit criteria for
two isometries to generate the same CSL. Moreover, we determine the number of
CSLs of a given index and encapsulate the result in a Dirichlet series
generating function.
