  We propose an explicit conjectural lift of the Neron-Severi Lie algebra of a
hyperk\"ahler variety $X$ of $K3^{[2]}$-type to the Chow ring of
correspondences ${\rm CH}^\ast(X \times X)$ in terms of a canonical lift of the
Beauville-Bogomolov class obtained by Markman. We give evidence for this
conjecture in the case of the Hilbert scheme of two points of a $K3$ surface
and in the case of the Fano variety of lines of a very general cubic fourfold.
Moreover, we show that the Fourier decomposition of the Chow ring of $X$ of
Shen and Vial agrees with the eigenspace decomposition of a canonical lift of
the grading operator.
