How can $J_1^2, J_2^2, J_{1z}, J_{2z}$ commute mutually? I'm reading through J. J. Sakurai's Modern Quantum Mechanics book and currently looking at the "Angular-momentum addition" part. 
Here, it says you have two options and that one option is to construct simultaneous eigenket $\vert j_1j_2;m_1m_2\rangle$ of $J_1^2, J_2^2, J_{1z}, J_{2z}$ since the four operators commute with each other. I understand that $J_1^2$ and $J_{1z}$ commute, but I'm not sure how $J_1^2$ and $J_{2z}$ can commute intuitively.
"commute" means that one can measure both at once right? But total angular momentum of spin 1 and angular momentum of spin 2 are independent.
Where am I wrong here?
 A: From your statement "measure both at once", it seems you've misunderstood what's meant by "simultaneous measurement".  It does not mean that you can run a single experiment to measure both "independent" values together.  Rather, it means that in principle, you can measure one quantity without "ruining" the results of measurement the other, so both quantities can be obtained.
A: Technically when two operators $A$ and $B$ commute it means that $AB = BA$, but from a physical standpoint yes it means that both observables can be measured simultaneously, and in that respect you kinda answered your own question. They're independent so they commute. For an analogy think of the spin of the electron in a hydrogen atom and its angular momentum about the CM, or think of the spins of two separate electrons. You can simultaneously measure these two quantities, why should they have anything to do with each other? Using the definition of commutativity it's a simple exercise to verify this: just operate both $J_1^2$ and $J_{2z}$ on the kets you've written and see what you get.
