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The Sagittarius dwarf spheroidal (Sgr) is a dissolving galaxy being tidallydisrupted by the Milky Way (MW). Its stellar stream still poses seriousmodelling challenges, which hinders our ability to use it effectively as aprobe of the MW gravitational potential. Our goal is to construct the bestpossible sample of stars with which we can advance our understanding of theSgr-MW interaction, focusing on the characterisation of the bifurcations. Weimprove on previous methods based on the use of the wavelet transform tosystematically search for the kinematic signature of the Sgr stream in the Gaiadata. We refine our selection via the use of a clustering algorithm on thestatistical properties of the colour-magnitude diagrams. Our final samplecontains > 700k candidate stars, 3x larger than previous samples. We have beenable to detect the bifurcation of the stream in both the northern and southernhemispheres, requiring four branches to fully describe this system. We presentthe detailed proper motion distribution of the trailing arm as a function ofLambda showing the presence of a sharp edge (on the side of the small propermotions) beyond which there are no Sgr stars. We also characterise thecorrelation between kinematics and distance. The chemical analysis of oursample shows a significant difference between the faint and bright branches. Weprovide analytical descriptions for the proper motion trends as well as for thesky distribution of the four branches of the stream. We interpret thebifurcations as the misaligned overlap of the material stripped at theantepenultimate pericentre (faint branches) with the stars ejected at thepenultimate pericentre (bright branch). The source of this misalignment isstill unknown but we argue that models with some internal rotation in theprogenitor, at least during the time of stripping of the stars in the faintbranches, are worth exploring.

The Sagittarius stream in Gaia eDR3 and the origin of the bifurcations