Euler's Four-Square Identity

Identity

The product of the sum of the four squares is itself, the sum of four squares.

Mathematically, for any eight complex numbers $x_1,x_2, x_3, x_4, y_1, y_2, y_3, y_4$ , we must have $(x_1^2+ x_2^2 + x_3 ^2 + x_4^2)(y_1^2+y_2^2+y_3^2+y_4^2)$ $=(x_1y_1+x_2y_2+x_3y_3+x_4y_4)^2$ $+(x_1y_2-x_2y_1+x_3y_4-x_4y_3)^2$ $+(x_1y_3-x_3y_1+x_4y_2-x_2y_4)^2$ $+(x_1y_4-x_4y_1 + x_2y_3 - x_3y_2)^2.$

This is Euler's Four-Square Identity.

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Euler's Four-Square Identity

Identity

Proof