We present transition path sampling simulations of the nucleation of the demixing transition in a binary symmetric Lennard-Jones fluid. In this system the demixing transition takes place between two phases of the same density but different compositions. The appropriateness of the reaction coordinate of classical nucleation theory is examined. Using paths harvested with transition path sampling, we investigate the nucleation mechanism and analyze the properties of critical nuclei obtained by determining the transition state ensemble. Our simulations show that despite the fact that the densities of the coexisting phases are equal, the density of the growing cluster plays a crucial role in the nucleation process: nucleation tends to proceed either via small, compact clusters with densities below that of the metastable fluid or via large clusters with even lower densities.