Given an element $g$ in matrix Lie group $SO(3)$ such that $g$ commutes with all elements in $SO(3)$ that represents rotation around $z-$axis. Are such element $g$ also forced to be a rotation around $z-$axis? Are there any other possibilities? Calculation argument should work, but I want a "clever" argument.
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It's not difficult to see that $g$ must send the $z$-axis to the $z$-axis (consider that $g$ commutes with the $90^\circ$ rotation about the $z$ axis). So if $g$ is not a rotation about the $z$ axis, there is only one option: it flips the $z$ axis. However, such an operation again cannot commute with a $90^\circ$ rotation about the $z$ axis.
Arthur
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