:::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-½ 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