  It is known that summations over Weyl groups of Lie algebras is a problem
which enters in many areas of physics as well as in mathematics. For this, a
method which we would like to call {\bf permutation weights} has been
previously proposed for pairs $(G_N, A_{N-1})$ of Lie algebras. It is now
extended for $(E_7, A_7)$ and also $(E_8, A_8)$. It is clear that these are the
most non-trivial ones and hence deserve studying separately.
  In order to obtain the results of these summations in practice, it is shown
that some simplifications occur in the method which is previously proposed for
pairs $(A_N, A_{N-1})$ in an unpublished work.
