  We show that the cube of the Hopf map $\eta$ maps to zero under the Hurewicz
map for all fixed points of all norms to cyclic $2$-groups of the
Landweber-Araki Real bordism spectrum. Using that the slice spectral sequence
is a spectral sequence of Mackey functors, we compute the relevant portion of
the homotopy groups of these fixed points, showing that multiplication by $4$
annihilates $\pi_{3}$.
