Pervasive ellipticity in locally compact groups

A topological group is (openly) almost-elliptic if it contains a(n open) dense subset of elements generating relatively-compact cyclic subgroups. We classify the (openly) almost-elliptic connected locally compact groups as precisely those with compact maximal semisimple quotient and whose maximal compact subgroups act trivial-weight-freely on the Euclidean quotients of the closed derived series' successive layers. In particular, an extension of a compact connected group K by a connected, simply-connected solvable Lie group L is (openly) almost-elliptic precisely when the weights of the K-action on Lie(L) afforded by the extension are all non-trivial.