  We show that the direct sum of an odd number of matrices
$$C=\left(\begin{array}{cccc} 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0\\ 0&0&1&1
\end{array}\right)$$ cannot be a sum $P+Q$ of matrices over $\mathbb{F}_2$
satisfying $P^2=P$ and $Q^3=O$.
