Consider a matrix $X\equiv\left[\begin{matrix}Z & W\end{matrix}\right]$, with $Z^{\intercal}P_{W}Z$ and $W^{\intercal}P_{Z}W$ nonsingular, where $P_{\bullet}\equiv \bullet(\bullet^{\intercal}\bullet)^{-1}\bullet^\intercal$.
Intuitively, I'd say that $P_{X}P_{Z}=P_{Z}$ since $\mathrm{col}Z\subset \mathrm{colX}$, yet I can't show that from straight arithmetics. Next is my work.
Denote $M_\bullet\equiv I-P_{\bullet}$. Then:
$ P_{X}P_{Z} =\left[\begin{matrix}Z & W\end{matrix}\right]\left[\begin{matrix}Z^{\intercal}Z & Z^{\intercal}W\\ W^{\intercal}Z & W^{\intercal}W \end{matrix}\right]^{-1}\left[\begin{matrix}Z^{\intercal}\\ W^{\intercal}P_{Z} \end{matrix}\right] =\left[\begin{matrix}Z & W\end{matrix}\right]\left[\begin{matrix}\left(Z^{\intercal}Z-Z^{\intercal}P_{W}Z\right)^{-1} & -\left(Z^{\intercal}Z\right)^{-1}Z^{\intercal}W\left(W^{\intercal}W-W^{\intercal}P_{Z}W\right)^{-1}\\ -\left(W^{\intercal}W\right)^{-1}W^{\intercal}Z\left(Z^{\intercal}Z-Z^{\intercal}P_{W}Z\right)^{-1} & \left(W^{\intercal}W-W^{\intercal}P_{Z}W\right)^{-1} \end{matrix}\right]\left[\begin{matrix}Z^{\intercal}\\ W^{\intercal}P_{Z} \end{matrix}\right] =Z\left(Z^{\intercal}Z-Z^{\intercal}P_{W}Z\right)^{-1}Z^{\intercal}-P_{Z}W\left(W^{\intercal}W-W^{\intercal}P_{Z}W\right)^{-1}W^{\intercal}P_{Z}-P_{W}Z\left(Z^{\intercal}Z-Z^{\intercal}P_{W}Z\right)^{-1}Z^{\intercal}+W\left(W^{\intercal}W-W^{\intercal}P_{Z}W\right)^{-1}W^{\intercal}P_{Z} =Z\left(Z^{\intercal}M_{W}Z\right)^{-1}Z^{\intercal}-P_{Z}W\left(W^{\intercal}M_{Z}W\right)^{-1}W^{\intercal}P_{Z}-P_{W}Z\left(Z^{\intercal}M_{W}Z\right)^{-1}Z^{\intercal}+W\left(W^{\intercal}M_{Z}W\right)^{-1}W^{\intercal}P_{Z} =M_{W}Z\left(Z^{\intercal}M_{W}Z\right)^{-1}Z^{\intercal}+M_{Z}W\left(W^{\intercal}M_{Z}W\right)^{-1}W^{\intercal}P_{Z}$
I am also thinking that the right-hand term in the last line is $0$, but I can't show that either. What am I missing?
Note that $Z = X \begin{pmatrix} I \\ \mathbf{0} \end{pmatrix}$ where $I$ is an identity matrix of the same order as $Z$ and $\mathbf{0}$ is a $0$ matrix of the same order as $W$.
So
$ \begin{align} P_XP_Z &= X(X^\top X)^{-1}X^T Z(Z^\top Z)^{-1} Z^\top\\ &= X(X^\top X)^{-1}X^TX\begin{pmatrix} I \\ \mathbf{0} \end{pmatrix} (Z^\top Z)^{-1} Z^\top\\ &=X\begin{pmatrix} I \\ \mathbf{0} \end{pmatrix} (Z^\top Z)^{-1} Z^\top = Z(Z^\top Z)^{-1} Z^\top = P_Z\\ \end{align} $