The Extended Lie bracket operator is introduced for the analysis and control of nonlinear time-delay systems. This tool is used to characterize the integrability conditions of a given submodule. The obtained results have two fundamental outcomes. First, they define the necessary and sufficient conditions under which a given set of nonlinear one-forms in the n-dimensional delayed variables x(t),..., x(t-sD) where D is constant, are integrable. Secondly, they set the basis for the extension to this context of the geometric approach used for delay-free systems.