  To a compact Lie group $G$ one can associate a space $E(2,G)$ akin to the
poset of cosets of abelian subgroups of a discrete group. The space $E(2,G)$
was introduced by Adem, F. Cohen and Torres-Giese, and subsequently studied by
Adem and G\'omez, and other authors. In this short note, we prove that $G$ is
abelian if and only if $\pi_i(E(2,G))=0$ for $i=1,2,4$. This is a Lie group
analogue of the fact that the poset of cosets of abelian subgroups of a
discrete group is simply--connected if and only if the group is abelian.
