  By computing the double-resonant Raman scattering cross-section completely
from first principles and including electron-electron interaction at the $GW$
level, we unravel the dominant contributions for the double-resonant 2D-mode in
bilayer graphene. We show that, in contrast to previous works, the so-called
inner processes are dominant and that the 2D-mode lineshape is described by
three dominant resonances around the $K$ point. We show that the splitting of
the TO phonon branch in $\Gamma-K$ direction, as large as 12 cm$^{-1}$ in $GW$
approximation, is of great importance for a thorough description of the 2D-mode
lineshape. Finally, we present a method to extract the TO phonon splitting and
the splitting of the electronic bands from experimental data.
