For any $n \times p$ matrix $B \in \mathbb{R}^{n \times p}$ such that $B^{\top} B$ is invertible, define the projection matrix $P_B$ as $$ P_B = B (B^{\top} B)^{-1} B^{\top} \in \mathbb{R}^{n \times n}. $$ Now consider four matrices as follows $$ B \in \mathbb{R}^{n \times p_1}, \qquad C \in \mathbb{R}^{n \times p_2}, \qquad C \in \mathbb{R}^{n \times p_3}. $$ What I want to do is find the relationship between $P_{[B, C, D]}$ with $[B, C, D] \in \mathbb{R}^{n \times (p_1 + p_2 + p_3)}$ and $P_{[B, D]}$ with $[B, D] \in \mathbb{R}^{n \times (p_1 + p_3)}$. I wonder whether there exists some simple expressions for the difference $$ P_{[B, C, D]} - P_{[B, D]}? $$ Here the assumptions we have is the inverses of matrices in $P_{[B, C, D]}$ and $P_{[B, D]}$ exist. Could anyone help me? Thanks in advance.
PS: I found some formula of Inverse of a 3x3 block matrix, see here, and tried several times. I wonder there exists another approach to solve this problem more easily.