  It is well known that anomaly cancellations for D_16 Lie algebra are at the
root of the first string revolution. For E_8 Lie algebra, cancellation of
anomalies is the principal fact leading to the existence of heterotic string.
They are in fact nothing but the 6th order cohomologies of corresponding Lie
algebras. Beyond 6th order, the calculations seem to require special care and
it could be that their study will be worthwhile in the light of developments of
the second string revolution.
  As we have shown in a recent article, for A_N Lie algebras, there is a method
which are based on the calculations of Casimir eigenvalues. This is extended to
E_8 Lie algebra in the present article. In the generality of any irreducible
representation of E_8 Lie algebra, we consider 8th and 12th order cohomologies
while emphasizing the diversities between the two. It is seen that one can
respectively define 2 and 8 basic invariant polinomials in terms of which 8th
and 12th order Casimir eigenvalues are always expressed as linear
superpositions. All these can be easily investigated because each one of these
invariant polinimials gives us a linear equation to calculate E_8 weight
multiplicities. Our results beyond order 12 are not included here because they
get more complicated though share the same characteristic properties with 12th
order calculations.
