  For an untwisted affine Lie algebra we prove an embedding of any higher level
Demazure module into a tensor product of lower level Demazure modules (e.g.
level one in type A) which becomes in the limit (for anti-dominant weights) the
well-known embedding of finite-dimensional irreducible modules of the
underlying simple Lie algebra into the tensor product of fundamental modules.
To achieve this goal, we first simplify the presentation of these modules
extending the results of \cite{CV13} in the $\mathfrak{g}$-stable case. As an
application, we propose a crystal theoretic way to find classical
decompositions with respect to a maximal semi-simple Lie subalgebra by
identifying the Demazure crystal as a connected component in the corresponding
tensor product of crystals.
