  We introduce a class of cycles, called nondegenerate, strictly decomposable
cycles, and show that the image of each cycle in this class under the refined
cycle map to an extension group in the derived category of arithmetic mixed
Hodge structures does not vanish. This class contains certain cycles in the
kernel of the Abel-Jacobi map. The construction gives a refinement of Nori's
argument in the case of a self-product of a curve. As an application, we show
that a higher cycle which is not annihilated by the reduced higher Abel-Jacobi
map produces uncountably many indecomposable higher cycles on the product with
a variety having a nonzero global 1-form.
