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- For a spin 1/2 particle, why does Sx, Sy and Sz don't share the complete eigenspace even though all of them commute with S^2

We know that S

^{2}commutes with S_{z}and so they share their eigenspace. Now since S^{2}also commutes with S_{x}, as per my understanding, the eigenvectors of S^{2}and S_{z}should also be the eigenvectors of S_{x}. But since the paulic matrices σ_{x}and σ_{y}are not diagonlized in the eigenbasis of S^{2}, it is clear that S^{2}and S_{x}don't share their eigenspace even though they commute with each other. How is that possible? what am i missing?