  The surface of lines in a cubic fourfold intersecting a fixed line splits
motivically into two parts, one of which resembles a K3 surface. We define the
analogue of the Beauville-Voisin class and study the push-forward map to the
Fano variety of all lines with respect to the natural splitting of the
Bloch-Beilinson filtration introduced by Mingmin Shen and Charles Vial.
