On the motif of a cubic hypersurface

We consider a nonsingular cubic hypersurface $V$ in $\mathbf P^4$. We prove that the motif $\widetilde V$ can be expressed by means of the Tate motif and the motif $(Y,\frac12\operatorname{id}-\frac12c(\gamma))$, where $Y$ is the curve of straight lines on $V$ that pass through a fixed line $l_0\subset V$ and $\gamma$ is an automorphism of $Y$ that leaves no line coplanar with $l_0$ fixed.