  In the first part of this paper we will prove the Voevodsky's nilpotence
conjecture for smooth cubic fourfolds and ordinary generic Gushel-Mukai
fourfolds. Then, making use of noncommutative motives, we will prove the
Voevodsky's nilpotence conjecture for generic Gushel-Mukai fourfolds containing
a $\tau$-plane $\G(2,3)$ and for ordinary Gushel-Mukai fourfolds containing a
quintic del Pezzo surface.
