  We provide a combinatorial study of split matroids, a class that was
motivated by the study of matroid polytopes from a tropical geometry point of
view. A nice feature of split matroids is that they generalize paving matroids,
while being closed under duality and taking minors. Furthermore, these matroids
proved to be useful in giving exact asymptotic bounds for the dimension of the
Dressian, and also implied new results on the rays of the tropical
Grassmannians.
  In the present paper, we introduce the notion of elementary split matroids, a
subclass of split matroids that contains all connected split matroids. We give
a hypergraph characterization of elementary split matroids in terms of
independent sets, and show that the proposed class is closed not only under
duality and taking minors but also truncation. We further show that, in
contrast to split matroids, the proposed class can be characterized by a single
forbidden minor. As an application, we provide a complete list of binary split
matroids.
