What is the meaning of the commutation relations (spin $1/2$ particles)?

I've often seen spin 1/2 commutation rules as a principle valid for every angular momentum. In some text books there is a derivation from symmetries principles. My question is, if I have a spin $$1/2$$ particle, which measurements should I perform to know which are the commutation rules of $$S_x, S_y, S_z$$?

Commutation rules do not follow from measurements, but from comparing two sequence of rotations about different axes with inverse angles: \begin{align} 1-\lambda^2[L_x,L_y]&\approx e^{i \lambda L_x}e^{i\lambda L_y} e^{-i\lambda L_x} e^{-i\lambda L_y}\, ,\\ &= \left(R_x(\lambda)R_y(\lambda)\right)\left(R_x(-\lambda)R_y(-\lambda)\right) \end{align}
It may also be possible to obtain information about the average value of a commutator from the uncertainty relation \begin{align} (\Delta_{\vert\psi\rangle}L_x)(\Delta_{\vert\psi\rangle}L_y)=\frac{1}{2}\vert \langle \psi\vert [L_x,L_y]\vert \psi\rangle\vert \end{align} but this relation is state dependent so it’s not terribly useful.
• ... and no I don’t think you can do measurement to infer commutation relations v.g. take an eigenstate of $L_x$ then $\langle L_y\rangle=\langle L_z\rangle=0$ so what can one infer from that? – ZeroTheHero Aug 16 '20 at 15:15