  Given two orientable, cusped hyperbolic 3-manifolds containing certain
thrice-punctured spheres, Adams gave a diagrammatic definition for a third such
manifold, their belted sum. Fully augmented links, or FALs, are hyperbolic
links constructed by augmenting a link diagram. This work considers belted sum
decompositions in which all manifolds involved are FAL complements. To do so,
we provide explicit classifications of thrice punctured spheres in FAL
complements, making them easily recognizable. These classifications are used to
characterize belted sum prime FALs geometrically, combinatorially and
diagrammatically. Finally we prove that, in the context of belted sums, every
FAL complement canonically decomposes into FALs which are either prime or
two-fold covers of the Whitehead link.
