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. L. Griess' paper "[Elementary abelian $p$-subgroups of algebraic groups](https://doi.org/10.1007/BF00150757)", Andersen–Grodal–Möller–Viruel "[The classification of $p$-compact groups for $p$ odd](https://doi.org/10.4007/annals.2008.167.95)" and J. Yu, "[Elementary abelian $2$-subgroups of compact Lie groups](https://doi.org/10.1007/s10711-012-9813-2)". These paper also contain a detailed history of the subject.