Synthesis of a reaction cluster is required when the yield of a desired product from a single chemical reaction is too low for practical application. For a desired overall reaction, there may be a large number of industrially operable clusters, and it is our goal to find the optimal one among these. In this paper, a lexicographic search algorithm is proposed, which identifies all potential clusters that satisfy user specified conditions for a given set of chemical species. As a case study, this algorithm is applied to synthesis of reaction clusters for the decomposition of water to its constitutive elements. Successful execution of this reaction will provide us with sustainable clean energy. As a result of this study, various potential clusters have been synthesized at relatively low operation temperatures. These clusters can be more thoroughly investigated for potential side reactions, kinetics, and other concerns. This methodology can be applied to any desired overall reaction. Therefore, it is expected to contribute to design of economic, safe, and environmentally benign chemical processes.