A procedure for finding locally the linearizing output of a single input nonlinear affine system is proposed. It relies on successive integrations of one-dimensional distributions and projections along these submanifolds. The algorithm proceeds recursively reducing the dimension one by one of both the number of coordinates and the number of vector fields, until the solution is obtained. A variant of the algorithm is also proposed, which does not require the computation of the full initial distribution. The proof of convergence of this second algorithm shows the importance of a new anti- symmetrical product. Besides providing a new insight into the involutivity condition, the algorithm can lead to a simple way of integrating the system of partial differential equations defining the linearizing output.