I have been reading Modern Quantum Mechanics by J.J.Sakurai. Under the chapter Quantum Dynamics, the author says if an observable $A$ initially commutes with the Hamiltonian operator $H$, then it remain so for all later times. Also, he mentions that this is how an observable commuting with $H$ remains a constant of motion.
Question: What's special about $H$? Are the above statements valid if we used some other operator than $H$? For example, can we say that if the observables $A$ and $B$ are compatible, then they remain so for all other time?