  Consider an external product of a higher cycle and a usual cycle which is
algebraically equivalent to zero. Assume there exists an algebraically closed
subfield k such that the higher cycle and its ambient variety are defined over
k, but the image of the usual cycle by the Abel-Jacobi map is not. Then we
prove that the external product is nonzero if the image of the higher cycle by
the cycle map to the reduced Deligne cohomology does not vanish. We also give
examples of indecomposable higher cycles on even dimensional hypersurfaces of
degree at least four in a projective space such that the last condition is
satisfied.
