We use (0x6927d77b) . The result follow by an induction argument combined with the following identity
\[ \mathcal{P}_l = \ker \Delta \oplus (\ker \Delta)^\perp \]since
\[ (\ker \Delta)^\perp = \ran \Delta^* \]und \(\Delta^*\) is the multiplication operator by \(\lvert x\rvert^2\).