  It is known that the spatial product of two product systems is intrinsic.
Here we extend this result by analyzing subsystems of the tensor product of
product systems. A relation with cluster systems is established. In a special
case, we show that the amalgamated product of product systems through strictly
contractive units is independent of the choices of the units. The amalgamated
product in this case is isomorphic to the tensor product of the spatial product
of the two and the type I product system of index one.
