Euler's four-square identity

Note that each quadrant reduces to a version of Euler's four-square identity:

Euler's four-square identity is an analogous identity involving four squares instead of two that is related to quaternions.

(This is essentially Euler's four-square identity.)

Similar statements are true for quaternions (Euler's four-square identity), complex numbers (the Brahmagupta-Fibonacci two-square identity) and real numbers.

(known as the Brahmagupta-Fibonacci identity), and also Euler's four-square identity and Degen's eight-square identity.

Euler's four-square identity or theorem, the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares.

Important precursors to this work included Euler's four-square identity (1748) and Olinde Rodrigues' parameterization of general rotations by four parameters (1840), but neither of these writers treated the four-parameter rotations as an algebra.