Splitting of abelian varieties in motivic stable homotopy category

In this paper, we discuss the motivic stable homotopy type of abelian varieties. For an abelian variety over a field k with a rational point, it always splits off a top-dimensional cell in motivic stable homotopy category SH(k). Let k = R, there is a concrete splitting which is determined by the motive of X and the real points X(R) in SH(R)Q. We will also discuss this splitting from a viewpoint of the Chow-Witt correspondences.