A "not well understood" proof, do you know if one knows by now the conceptual background?: http://www.math.u-bordeaux1.fr/~cohen/sum4cub.ps
There is a paper by Philippe Revoy that analyses and expands on this result.
Its abstract reads (the paper itself is in French):
The paper itself seems to be freely available on seals.ch, the link below should take you there directly:
Ph. Revoy, Sur les sommes de quatre cubes, Enseign. Math 29 (1983), 209-220