Bell's theorem for systems more complicated than two qubits faces a hidden,as yet undiscussed, problem. One of the methods to derive Bell's inequalitiesis to assume existence of joint probability distribution for measurementresults for all settings in the given experiment. However for spins-1, onefaces the problem that eigenvalues of observables do not allow a consistentalgebra if one allows all possible settings on each side (Bell 1966contradiction), or some specific sets (leading to a Kochen-Specker 1967contradiction). We show here that by choosing special set of settings whichnever lead to inconsistent algebra of eigenvalues, one can still derivemultisetting Bell inequalities, and that they are robustly violated. Violationfactors increase with the number of subsystems. The inequalities involve onlyspin observables, we do not allow all possible qutrit observables, still theviolations are strong.
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