While reading the paper From Groups to Groupoids: a survey by Ronald Brown, I found this intriguing remark:
[...] the classification of real vector spaces with $p$ endomorphisms is interesting for $p = 1$, difficult for $p = 2$, and unsolved for $p = 3$. (I am grateful to A. Heller for this trenchant expression of view.)
This seems a really interesting problem, I looked for papers and discussions regarding the case $p = 2$ but I couldn't find any.
Perhaps someone here can point me to the right keywords or authors I have missed?