Summary: ``Through a geometric understanding of the creation, cap, annihilation and cup operators for {\it ADE} graphs in $SU(3)$ we propose the first steps towards an algorithm that would allow one to write an arbitrary elementary path as an ordered combination of creation and cap operators acting upon an essential path. We propose a sketch of a proof and use our proposal for some examples for the $A_2$ and $E_5$ graphs of the $SU(3)$ family. Attaining this decomposition is an important step in obtaining the path formulation of the quantum Algebra of a modular invariant RCFT.''