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We investigate ways to produce the bifurcation observed in the stellar streamof the Sagittarius dwarf galaxy (Sgr). Our method consists in running $N$-bodysimulations of Sgr falling into the Milky Way for the last 3~Gyr, with addedtest particles on disk orbits that span a wide range of initial positions,energies, and angular momenta. We find that particles that end up in the faintbranch are predominantly high angular momentum particles that can all originatefrom a single plane within the progenitor, nearly perpendicular both to theorbital plane of the progenitor and to the Milky Way stellar disk. Theiroriginal configuration at the start of the simulation corresponds to spiralfeatures already present 3~Gyr ago, which could be, e.g., the result of adisk-like component being tidally perturbed, or the tidal tails of a satellitebeing disrupted within Sgr. We then run a simulation including the self-gravityof this disky component. Despite the remaining ambiguity of its origin, thisdisk component of the Sgr dwarf with spiral over-densities provides a firststep towards a working model to reproduce the observed faint branch of thebifurcated Sgr stream.

Revisiting a disky origin for the faint branch of the Sagittarius stellar stream