The kinematic analysis and synthesis of parallel mechanisms, particularly those with lower mobility or redundancy, present significant challenges in robotics and mechanical engineering. Traditional approaches often struggle to precisely define the task-relevant motion space when parasitic motions are present. This article introduces a novel method for identifying the "quotient manifold"—the geometric representation of the pure, task-relevant motion of a mechanism—by leveraging Chasles' decomposition models for finite displacements. The proposed methodology provides a systematic approach to characterize the global motion capabilities of "quotient mechanisms," which are designed to produce a specific set of motions while inherent parasitic motions are ignored or decoupled. We outline a theoretical framework rooted in Lie group theory and screw algebra, detailing the algorithm for representing finite displacements as screw motions and subsequently mapping them to identify the quotient manifold. Hypothetical results demonstrate the method's ability to accurately capture the intended motion space and distinguish it from parasitic motions. This approach offers enhanced clarity in kinematic analysis, provides a robust foundation for the type synthesis and optimal design of lower-mobility parallel mechanisms, and contributes to a deeper understanding of complex mechanical system behavior.