  The basis exchange axiom has been a driving force in the development of
matroid theory. However, the axiom gives only a local characterization of the
relation of bases, which is a major stumbling block to further progress, and
providing a global understanding of the structure of matroid bases is a
fundamental goal in matroid optimization.
  While studying the structure of symmetric exchanges, Gabow proposed the
problem that any pair of bases admits a sequence of symmetric exchanges. A
different extension of the exchange axiom was proposed by White, who
investigated the equivalence of compatible basis sequences. These conjectures
suggest that the family of bases of a matroid possesses much stronger
structural properties than we are aware of.
  In the present paper, we study the distance of basis pairs of a matroid in
terms of symmetric exchanges. In particular, we give a polynomial-time
algorithm that determines a shortest possible exchange sequence that transforms
a basis pair into another for split matroids, a class that was motivated by the
study of matroid polytopes from a tropical geometry point of view. As a
corollary, we verify the above mentioned long-standing conjectures for this
large class. Being a subclass of split matroids, our result settles the
conjectures for paving matroids as well.
