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  "body": "\n## Statement\n\nLet $a_1, \\dots, a_4$ and $b_1, \\dots, b_4$ be elements of a commutative ring. Then\n$$\\begin{aligned} (a_1^2 + a_2^2 + a_3^2 + a_4^2)(b_1^2 + b_2^2 + b_3^2 + b_4^2) \u0026= (a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4)^2 + (a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3)^2 \\\\ \u0026\\quad + (a_1 b_3 - a_2 b_4 + a_3 b_1 + a_4 b_2)^2 + (a_1 b_4 + a_2 b_3 - a_3 b_2 + a_4 b_1)^2. \\end{aligned}$$\n\n## Remarks\n\nSo the product of two sums of four squares is again a sum of four squares.\nThe signs come from the product of two quaternions: the identity says that the norm of quaternions is multiplicative.\nLagrange used it to reduce the [[four-square theorem|0KYYSMEVHH]] to the case of primes.\n\n## History\n\nEuler wrote about the identity in a letter to Goldbach of 4 May 1748.\n\n## Explanations\n\n### L1 human\n\nThis identity is the four-square analogue of the [[Brahmagupta–Fibonacci identity|0K2DMACEZ4]] for two squares.\nYou can check it by expanding both sides, since all the cross terms cancel.\n",
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