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