  The full root polytope of type $A$ is the convex hull of all pairwise
differences of the standard basis vectors which we represent by forward and
backward arrows. We completely classify all flag triangulations of this
polytope that are uniform in the sense that the edges may be described as a
function of the relative order of the indices of the four basis vectors
involved. These fifteen triangulations fall naturally into three classes: three
in the lex class, three in the revlex class and nine in the Simion class. We
also consider a refined face count where we distinguish between forward and
backward arrows. We prove the refined face counts only depend on the class of
the triangulations. The refined face generating functions are expressed in
terms of the Catalan and Delannoy generating functions and the modified Bessel
function of the first kind.
