Several issues in the toroidal angular momentum transport by the ion temperature gradient (ITG) turbulence are addressed in this work: the system size effect in the momentum transport, the symmetry breaking mechanism for the generation of the spontaneous rotation and the effect of the trapped electron dynamics. We find that the magnitude of the momentum flux scales with the system size according to the gyroBohm scaling, with no significant size effect on the radial structure of the perturbed toroidal angular momentum. The symmetry breaking due to the shear of the radial electric field is found to be a mechanism for generating the residual momentum flux. However, it is small compared with the momentum pinch term in the case of a finite background rotation. The trapped electrons in the ITG turbulence increase the intensity and modify the spectral properties of the electrostatic fluctuations, leading to the increase in the toroidal momentum pinch.