  In this paper we give new methods to construct zero divisors in A_n =R^(2^n)
the Cayley_Dickson algebras over the real numbers, for n larger than 4, and we
also relate the set of zero divisors in A_{n+1} with the Stiefel Manifold
V_{2^n -1,2} for n>3. We also introduce the notion of "Spectrum" (of a no zero
double pure element) wich synthesize the information regarding the structure of
the linear operators left and right multiplication by the element. We use this
as a main technical tool to construct the zero divisors.
