Algebraic cycles and intersections of 2 quadrics

A smooth intersection $Y$ of two quadrics in ${\mathbb{P}}^{2g+1}$ has Hodge level 1. We show that such varieties $Y$ have a multiplicative Chow-K"unneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of $Y$ injects into cohomology.