Previous works have shown that studying the characteristics of the Quadratic Assignment Problem (QAP) is a crucial step in gaining knowledge that can be used to design tailored meta-heuristic algorithms. One way to analyze the characteristics of the QAP is to decompose its objective function into a linear combination of orthogonal sub-functions that can be independently studied. In particular, this work focuses on a decomposition approach that has attracted considerable attention: The Elementary Landscape Decomposition (ELD).The main drawback of the ELD is that it does not allow an understandable characterization of what is being measured by each component of the decomposition. Thus, it turns out difficult to design new efficient meta-heuristic algorithms for the QAP based on the ELD. To address this issue, in this work, we delve deeper into the ELD by means of an additional decomposition of its elementary components. Conducted experiments show that the performed analysis may be used to explain the behaviour of ELD-based methods, providing critical information about their potential applications.