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Resolving a conjecture of von Neumann, Ogata's theorem in arXiv:1111.5933showed the highly nontrivial result that arbitrarily many matricescorresponding to macroscopic observables with $N$ sites and a fixed sitedimension $d$ are asymptotically nearby commuting observables as $N \to\infty$. In this paper, we develop a method to construct nearby commutingmatrices for normalized highly reducible representations of $su(2)$ whosemultiplicities of irreducible subrepresentations exhibit a certainmonotonically decreasing behavior. We then provide a constructive proof ofOgata's theorem for site dimension $d=2$ with explicit estimates for how closethe nearby observables are. Moreover, motivated by the application totime-reversal symmetry explored in arXiv:1012.3494, our construction has theproperty that real macroscopic observables are asymptotically nearby realcommuting observables.

Constructing Nearby Commuting Matrices for Reducible Representations of $su(2)$ with an Application to Ogata's Theorem