How to determine (say up to conjugacy) elementary $p$-subgroups of a compact Lie group $G$?


Of course there are the $p$-subgroups of a maximal torus, and in the case $G=\mathrm{PU}\_p$, there is an interesting non-toral elementary $p$-subgroup considered by Vistoli in [this paper](https://arxiv.org/abs/math/0505052).


How many other cases are known? For example, how about $G=PU\_n$ where $n$ is not a prime?



Elementary $p$-subgroups of a compact Lie group
This is a topic with a long history going (at least) back to Borel, Serre, Steinberg and others. The existence of *non-toral* 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](https://doi.org/10.1007/BF00150757)", Andersen–Grodal–Møller–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.


