If $A$ and $B$ are two commuting observables on a Hilbert space of dimension $n$ say. So, $$A = \sum_{j \leq a} \lambda_j P_j $$
$$B = \sum_{i \leq b} \mu_i Q_i $$
$$I_n = \sum_{i \leq b} Q_j = \sum_{j \leq a} P_j = \sum_{j \leq a, i \leq b} P_jQ_i$$
and the $P$'s are mutually orthogonal, the $Q$ too. We know that the $P$'s and the $Q$'s are commuting and that for any $i$ there is at least one $j$ such that $Q_iP_j$ is not zero. How do we show that the number of non-zero products of the $P$'s and the $Q$'s is less than $n$ ?
There is proof using the decomposition in direct sum of the Hilbert space and getting a contradiction with the dimensions. But is there a proof purely algebraic, using only the properties of the projectors ?