How to write $M_{12}$ in terms of $X_1$ and $(I-\Bbb{P}_{X_1})X_{2.1}$?

Question

Let $X$ be a $n\times p$ data matrix from $N_p(0,\Sigma)$. $X=(\underbrace{X_1}_q\ \ | \ \ \underbrace{X_2}_{p-q})$

$M=X^tX\sim W_p(\Sigma,n)$ $$M=\left(\begin{matrix}\underbrace{M_{11}}_{q\times q} & M_{12}\\ M_{21} & \underbrace{M_{22}}_{p-q\times p-q}\end{matrix}\right)$$

Define $\Bbb{P}_{X_1}=I_{q\times q}-X_1M_{11}^{-1}X_1^t$ and $X_{2.1}=X_2-\Sigma_{21}\Sigma_{11}^{-1}X_1$

Question:

How to write $M_{12}$ in terms of $X_1$ and $(I-\Bbb{P}_{X_1})X_{2.1}$?

NB:

We know $\Bbb{P}_{X_1}\cdot X_{2.1}$ and $(I-\Bbb{P}_{X_1})\cdot X_{2.1}$ are statistically independent.