  Demazure crystals are subcrystals of highest weight irreducible
$\mathfrak{g}$-crystals. In this article, we study tensor products of a larger
class of subcrystals, called extremal, and give a local characterization for
exactly when the tensor product of Demazure crystals is extremal. We then show
that tensor products of Demazure crystals decompose into direct sums of
Demazure crystals if and only if the tensor product is extremal, thus providing
a sufficient and necessary local criterion for when the tensor product of
Demazure crystals is itself Demazure. As an application, we show that the
primary component in the tensor square of any Demazure crystal is always
Demazure.
