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.
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