Here’s a nice little math identity that mathematician Leonard Euler wrote down in a letter dated 1748. It states that the product of two numbers that are sums of four squares is itself the sum of four squares.
(a1b1-a2b2-a3b3-a4b4)2 + (a1b2+a2b1+a3b4-a4b3)2 + (a1b3+a3b1+a4b2-a2b4)2 + (a1b4+a4b1+a2b3-a3b2)2
It can be proved with elementary algebra or even by quaternions!
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