This paper presents a new and effective algorithm to compute a set of at outputs for nonlinear implicit systems described in the differential geometric framework of jets of infinite order. The proposed methodology is based on the necessary and sufficient conditions for differential flatness. A procedure is set up to manage the degrees of freedom left by the polynomial matrices approach. First, a reduced-order basis of the ideal of differential forms generated by the differentials of all possible Lie-Backlund isomorphisms is obtained by computing reduced order bases of the left nullspace of Ore polynomial matrices and the weak Popov form over a Weyl algebra. Then, the generalized Cartan moving frame, including additional strutural constraints, is used in order to characterize the strong closedness of the latter ideal.