This is a topic with a long history going (at least) back to Borel, Serre, Steinberg and others. The existence of <em>non-toral</em> elementary abelian $p$-subgroups (i.e.&nbsp;subgroups not contained in a maximal torus of $G$) is equivalent to $H_*(G;\mathbb{Z})$ having $p$-torsion. Newer results include R.&nbsp;L.&nbsp;Griess' paper "Elementary abelian $p$-subgroups of algebraic groups", Andersen-Grodal-Möller-Viruel "The classification of $p$-compact groups for $p$ odd" and J.&nbsp;Yu, "Elementary abelian $2$-subgroups of compact Lie groups". These paper also contain a detailed history of the subject.