The aim of the present paper is to introduce new mathematical tools for the analysis and control of nonlinear time-delay systems (NLTDS). An Extended Lie bracket operation equivalent to the Lie bracket operation for system without delays is introduced. It will be shown that this operation helps to characterize certain properties of a given submodule, such as nilpotency. This basic property is then used to define the conditions under which a given unimodular matrix represents a bicausal change of coordinates. The effectiveness of the proposed approach will be shown by solving an important basic problem: to characterize if a NLTDS is equivalent or not, to a Linear Time-Delay System by bicausal change of coordinates.