Geometry of numbers proof of Götzky's four-squares theorem. (Q1864870)

The author gives an elementary proof that every totally positive integer in \(\mathbb{Q}(\sqrt{5})\) is the sum of four squares (first proved by \textit{F. Götzky} [Math. Ann. 100, 411--437 (1928; JFM 54.0407.01)] with an exact count using Siegel modular functions). For \(\mathbb{Z}\), the author adapts Euler's original proof to Minkowski's theorem on a 4-sphere [see \textit{J. H. Grace}, The four square theorem, J. Lond. Math. Soc. 2, 3--8 (1927; JFM 53.0134.05)]. The new proof requires an 8-dimensional convex body bounded by the metric of the sum of the radial distances in each of two four-dimensional projections.