Assume that $\mathbb{R}^n$ is represented as the direct (but not necessarily orthogonal) sum $M_1 \dotplus M_2$ of two its subspaces $M_1$ and $M_2$. In particular, every $x \in \mathbb{R}^n$ can be represented in a unique way as $x = x_1 + x_2$ with some $x_j \in M_j$, and the mapping $P_j: x \mapsto x_j$ is called the (oblique) projector onto $M_j$ parallel to $M_{3-j}$. It is easy to show that $P_j$ satisfy the following properties: $P_1 + P_2 = I_n$, $P_j^2 = P_j$ and $P_1P_2 = P_2 P_1 = 0$.
Show that any matrix $P$ satisfying the relation $P^2 = P$ is a projector onto some subspace $L$ parallel to $M$, and identify these $L$ and $M$.