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