Let G be a semisimple real algebraic group defined over ${\mathbb {Q}}$ , $\Gamma $ be an arithmetic subgroup of G, and T be a maximal ${\mathbb {R}}$ -split torus. A trajectory in $G/\Gamma $ is divergent if eventually it leaves every compact subset. In some cases there is a finite collection of explicit algebraic data which accounts for the divergence. If this is the case, the divergent trajectory is called obvious. Given a closed cone in T, we study the existence of non-obvious divergent trajectories under its action in $G\kern-1pt{/}\kern-1pt\Gamma $ . We get a sufficient condition for the existence of a non-obvious divergence trajectory in the general case, and a full classification under the assumption that $\mathrm {rank}_{{\mathbb {Q}}}G=\mathrm {rank}_{{\mathbb {R}}}G=2$ .