  We settle the fusion product decomposition theorem for higher-level affine
Demazure modules for the cases
  $E^{(1)}_{6, 7, 8}, F^{(1)}_4$ and $E^{(2)}_{6}$, thus completing the main
theorems of Chari et al. (J. Algebra, 2016) and Kus et al. (Represent. Theory,
2016). We obtain a new combinatorial proof for the key fact, that was used in
Chari et al. (op cit.), to prove this decomposition theorem. We give a case
free uniform proof for this key fact.
