Commutative K-theory, a cohomology theory built from spaces of commuting matrices, has been explored in recent work of Adem, G{o}mez, Gritschacher, Lind, and Tillman. In this article, we use unstable methods to construct explicit representatives for the real commutative K-theory classes on surfaces. These classes arise from commutative O(2)-valued cocycles, and are analyzed via the point-wise inversion operation on commutative cocycles.
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