Absolute compatibility and poincar\'{e} sphere

In this paper, we introduce the notion of strict projections and prove that an absolutely compatible pair of strict elements in a von Neumann algebra $\mathcal{M}$ unitarily equivalent to the elements $ \left((p - x_0) \otimes I_2 \right) P_0 + (x_0 \otimes I_2) P$,$\left((p - x_0) \otimes I_2 \right) P_0 + (x_0 \otimes I_2) P’$of$M_2($\mathcal{M}$_0)$where$$\mathcal{M}$_0$ is an abelian von Neumann algebra, $x_0$ is a strict element of ${\mathcal{M}}_0^{+}$, $P_0=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\in M_2({\mathcal{M}}_0)$ and $P$ is a strict projection in $M_2({\mathcal{M}}_0)$. We also discuss the geometric form of this representation when $\mathcal{M}={\mathbb{M}}_2$.