Let $V=\mathbb C^n$ be a vector space with (linear) maps $P_1:V\rightarrow V$ and $P_2:V\rightarrow V$ that are projectors , i.e. they satisfy $P_i^2=P_i$.

It is not hard to understand the structure of such a pair. One could see that $P_i$'s can be simultaneously block diagonalized with blocks of size $2$. For this see, http://en.wikipedia.org/wiki/Principal_angles and also for example paper of Halmos with the title "two subspaces".

My question is about a generalization of this to pairs of $P_1, P_2$ where $P_i$'s satisfy a different (simple) algebraic relation say $P_i^3= P_i$.

I want to see whether they can be block-digonalized with blocks of size $O(1)$ independent of $n$.