Euler's four-square identity
In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares. Algebraic identity. For any pair of quadruples from a commutative ring, the following expressions are equal: Euler wrote about this identity in a letter dated May 4, 1748 to Goldbach (but he used a different sign convention from the above). It can be verified with elementary algebra. The identity was used by Lagrange to prove his four square theorem. More specifically, it implies that it  is sufficient to prove the theorem for prime numbers, after which the more general theorem follows. The sign convention used above corresponds to the signs obtained by multiplying two quaternions. Other sign conventions can be obtained by changing any formula_6 to formula_7, and/or any formula_8 to formula_9. If the formula_6 and formula_8 are real numbers, the identity expresses the fact that the absolute value of the product of two quaternions is equal to the product of their absolute values, in the same way that the Brahmagupta–Fibonacci two-square identity does for complex numbers. This property is the definitive feature  of composition algebras. Hurwitz's theorem states that an identity of form, where the formula_13 are bilinear functions of the formula_14 and formula_15 is possible only for "n" = 1, 2, 4, or 8. Proof of the identity using quaternions. Comment: The proof of Euler's four-square identity is by simple algebraic evaluation. Quaternions derive from the four-square identity, which can be written as the product of two inner products of 4-dimensional vectors, yielding again an inner product of 4-dimensional vectors: (a·a)(b·b) = (a*b)·(a*b). This defines the quaternion multiplication rule a*b, which simply reflects Euler's  identity, as well as the whole bunch of quaternion math! Quaternions are, so to say, the "square root" of the four-square identity. But let the proof go on: Let formula_16 and formula_17 be a pair of quaternions. Their quaternion conjugates are formula_18 and formula_19. Then and The product of these two is formula_22, where formula_23 is a real number, so it can commute with the quaternion formula_24, yielding No parentheses are necessary above, because quaternions associate. The conjugate of a product is equal to the commuted product of the conjugates of the product's factors, so where formula_27 is  the Hamilton product of formula_28 and formula_29: Then and Pfister's identity. Pfister found another square identity for any even power: If the formula_13 are just rational functions of one set of variables, so that each formula_13 has a denominator, then it is possible for all formula_45. Thus, another four-square identity is as follows: where formula_51 and formula_52 are given by Incidentally, the following identity is also true: