  In this paper, we consider the prehomogeneous vector space for pair of simple
algebras which are $k$-forms of the $D_4$ type and the $E_6$ type. We mainly
study the non-split cases. The main purpose of this paper is to determine the
principal parts of the global zeta functions associated with these spaces in
the non-split cases. We also give a description of the sets of rational orbits
of these spaces, which suggests the expected density theorems arising from the
properties of these zeta functions.
