  Fundamental elements are certain special elements of affine Weyl groups
introduced by Gortz, Haines, Kottwitz and Reuman. They play an important role
in the study of affine Deligne-Lusztig varieties. In this paper, we obtain
characterizations of the fundamental elements and their natural
generalizations. We also derive an inverse to a version of "Newton-Hodge
decomposition" in affine flag varieties. As an application, we obtain a
group-theoretic generalization of Oort's results on minimal p-divisible groups,
and we show that, in certain good reduction reduction of PEL Shimura datum,
each Newton stratum contains a minimal Ekedahl-Oort stratum. This generalizes a
result of Viehmann and Wedhorn.
