  We study the fan structure of Dressians $\Dr(d,n)$ and local Dressians
$\Dr(\cM)$ for a given matroid $\cM$. In particular we show that the fan
structure on $\Dr(\cM)$ given by the three term Pl\"ucker relations coincides
with the structure as a subfan of the secondary fan of the matroid polytope
$P(\cM)$. As a corollary, we have that a matroid subdivision is determined by
its 3-dimensional skeleton. We also prove that the Dressian of the sum of two
matroids is isomorphic to the product of the Dressians of the matroids. Finally
we focus on indecomposable matroids. We show that binary matroids are
indecomposable, and we provide a non-binary indecomposable matroid as a
counterexample for the converse.
