Prove that if
$\text{Proj}_{M_1}\text{Proj}_{M_2}= \text{Proj}_{M_2}\text{Proj}_{M_1}$ then
$\text{Pr}(\text{span}[M_1, M_2]) = \text{Pr}(M_1) + \text{Pr}(M_2) − \text{Pr}(M_1 \cap M_2)$.
In the case where the projection operators are non-commutative, I understand how to show that the above formula is actually false. However, I am unsure of how the commutativity of the projectors implies the above equation. I have included an image from the text to provide the definitions of $M_1$ and $M_2$.
It follows that $M_1$ is invariant subspace of operator $\text{Proj}_{M_2}$. Indeed, if $v \in M_1$ then $$\text{Proj}_{M_1}\text{Proj}_{M_2}v=\text{Proj}_{M_2}\text{Proj}_{M_1}v = \text{Proj}_{M_2}v,$$ so $$\text{Proj}_{M_1}(\text{Proj}_{M_2}v) = \text{Proj}_{M_2}v,$$ but this can happen only if $\text{Proj}_{M_2}v \in M_1$.
Similarly, it can be proved that $M_1^{\perp}$ is invariant for $\text{Proj}_{M_2}$, and also $M_2,M_2^{\perp}$ are invariant for $\text{Proj}_{M_1}$ by symmetry.
Since $M_1$ is invariant for $\text{Proj}_{M_2}$ and $\text{Proj}_{M_2}$ has two eigenspaces $M_2,M_2^{\perp}$, then $M_1$ can be split into $$ M_1 = (M_1 \cap M_2) \oplus (M_1 \cap M_2^{\perp}) $$ Similarly, $$ M_2 = (M_2 \cap M_1) \oplus (M_2 \cap M_1^{\perp}) $$
A more general case is discussed here https://quantumcomputing.stackexchange.com/a/6469/5870
