Euler Four-Square Identity -- from Wolfram MathWorld

The amazing polynomial identity communicated by Euler in a letter to Goldbach on April 12, 1749 (incorrectly given as April 15, 1705--before Euler was born--in Conway and Guy 1996, p. 232). The identity also follows from the fact that the norm of the product of two quaternions is the product of the norms (Conway and Guy 1996).

mathematics Archives - The Billy Lee Pontificator

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Polynomial Identity -- from Wolfram MathWorld

Euler Four-Square Identity -- from Wolfram MathWorld

Reply to @bus_.driver an example of an Euler brick and the

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Square (algebra) - Wikipedia

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Degen's Eight-Square Identity -- from Wolfram MathWorld

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PDF) Quarternions and the Four Square Theorem