The oxidation reaction of UO$_2$ into U$_3$O$_8$ is studied as a function of the crystalline distortion of interstitial oxygen clusters, named cuboctahedra, which appear in U$_4$O$_9$ and U$_3$O$_7$ intermediate phases. For that purpose, the refinement of α-U$_4$O$_9$ was performed because this phase undergoes a trigonal distortion from cubic β-U$_4$O$_9$ when the temperature is decreased. In α-U$_4$O$_9$, the cuboctahedra can be described as crumpled sheets taken from a fragment of U$_3$O$_8$ The manner by which the accumulation of crumpled sheets can lead to the formation of U$_3$O$_8$ is discussed.