  Kobayashi [Duke Math. J. (1992)] gave a necessary condition for the existence
of compact Clifford-Klein forms in terms of Cartan projections and non-compact
dimensions of reductive subgroups. We extend his method to non-reductive
subgroups, and give some examples of homogeneous spaces of reductive type that
do not admit compact Clifford-Klein forms by comparing Cartan projections and
non-compact dimensions of reductive subgroups with those of non-reductive
subgroups.
