Bell's inequality and operators' noncommutativity

Relations between Bell's inequality and noncommutativity of operators are discussed via the four operators involved in the Clauser et al inequality. The case of all operators commuting (i.e. the six commutators vanish) and the case of three out of the four operators mutually commuting (i.e. five commutators vanish) is shown to abide by the inequality. In the latter case a novel insight is unravelled. The Bell quantum bound (2 root 2) is obeyed for the case when four commutators vanish. The probabilistic upper limit of the inequality is reviewed and shown to be 4. In any theory based on Hilbert space, the upper limit is 2 root 3.