I am interested in discussing the applications of $A^1$-homotopy theory to the study of splitting behavior of projective modules developed by F. Morel, J. Fasel, A. Asok and their collaborators. I am also willing to present my results on $A^1$-homotopy invariance and related properties of non-stable $K_1$-functors associated to isotropic reductive groups (to appear in J. K-Theory). The latter results are potentially useful for extending the former ones to other principal G-bundles, G an isotropic reductive group.