Biochemical networks are models of biological functions and processes in biomedicine. Hierarchical decomposition simplifies complex biochemical networks by partitioning them into smaller blocks (modules), facilitating computationally intensive analyses and providing deeper insights into cellular processes and regulatory mechanisms. We introduce a novel algorithm for the hierarchical decomposition of large-scale biochemical systems. By using causality and information flow as organizing principles, our approach combines strongly connected components with $r$-causality to identify and structure manageable network blocks. Benchmarking against a comprehensive database of biochemical reaction networks demonstrates the computational efficiency and scalability of our algorithm. To ensure broad applicability, we integrate our algorithm into tools that support standardized Systems Biology Markup Language (SBML) formats, facilitating its use in biochemical modeling workflows.