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In an analogous construction as by Euler for 4x4 matrices, a parametrizationof 8x8 magic squares of squares with orthogonal rows is shown to be obtainableby extending the quaternionic method, as shown by Hurwitz, to octonions, butnot possible to be carried even further in the Cayley-Dickson constructionseries.

A parametrization of 8x8 magic squares of squares through octonionic multiplication