[UPSC CSE 2023] Angular momentum, spin etc (Q252, 15M)

An operator P describing the interaction of two spin \frac{1}{2} particles is P = a + b \vec{\sigma}_1 \cdot \vec{\sigma}_2, where a and b are constants, and \vec{\sigma}_1 and \vec{\sigma}_2 are Pauli matrices of the two spins. The total spin angular momentum \vec{S} = \vec{S}_1 + \vec{S}_2 = \frac{1}{2} \hbar (\vec{\sigma}_1 + \vec{\sigma}_2). Show that P, S^2 and S_z can be measured simultaneously.