Euler's four-square identity

<math>(a_1^2+a_2^2+a_3^2+a_4^2)(b_1^2+b_2^2+b_3^2+b_4^2)\,</math>

<math>=(a_1 b_1-a_2 b_2 - a_3 b_3 - a_4 b_4)^2 + (a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3)^2\,</math>

<math>+\,(a_1 b_3 - a_2 b_4 + a_3 b_1 + a_4 b_2)^2 + (a_1 b_4 + a_2 b_3 - a_3 b_2 + a_4 b_1)^2\,</math>

Euler wrote about this identity in 1750. It can be proven with elementary algebra and holds in every commutative ring. If the as and bs are real numbers, a more elegant proof is available: the identity expresses the fact that the absolute value of the product of two quaternions is equal to the product of their absolute values.

The identity was used by Lagrange to prove his four square theorem[?].