Large annihilators in Cayley-Dickson algebras

Cayley-Dickson algebras are nonassociative R-algebras that generalize the well-known algebras R, C, H, and O. We study zero-divisors in these algebras. In particular, we show that the annihilator of any element of the 2(n)-dimensional Cayley-Dickson algebra has dimension at most 2(n) - 4n + 4. Moreover, every multiple of 4 between 0 and this upper bound occurs as the dimension of some annihilator. Although a complete description of zero-divisors seems to be out of reach, we can describe precisely the elements whose annihilators have dimension 2(n) - 4n + 4.