We examine the $N$-partite quaternion and octonion dichotomic Bell inequalities derived by Vogel and Shchukin utilizing the square identities of Euler and Degen, which apply to experiments with $M$ settings at each spatially separated site. We reveal these inequalities to be violated by the Greenberger-Horne-Zeilinger state, for $N\ge 3$ and $2\le M\le 8$. Violations are robust with respect to loss, the threshold detection efficiency being $\eta >2^{\frac{2}{N}-1}$ for all $M$, implying violations for efficiencies as low as $\eta \sim 50\%$ at each site, as $N\to \infty$.

• Received 9 May 2015

DOI:https://doi.org/10.1103/PhysRevA.92.032118