When you write and you are presupposing that there are two states that are spin 1 2 i e that are eigenvalues of S 2 and that are also eigenvalues of S z Which means that you are already presupposing that S 2 and S z commute at least in the spin 1 2 space So it seems to me that it is inappropriate to use this kind of argument as a basis for proving that S 2 and S z commute Of course topics like Spin and Pauli matrices are important it might be worth saying more about that somewhere But in this article I think it should be put into the larger context i e discussing not only the spin 1 2 hilbert space but also spin 0 spin 1 etc Not to mention that real systems may be in more complicated spaces like a molecule can be in the direct sum of a spin 2 and a spin 4 space or an electron would be in the direct product of a spin 1 2 space and the space of complex functions on R 3 D