Euler four-square identityEulerFourSquareIdentity2002-04-16 18:22:502003-08-22 05:18:13Theorem% this is the default PlanetMath preamble. as your knowledge
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% define commands hereThe Euler four-square identity simply states that
$(x_1^2 + x_2^2 + x_3^2 + x_4^2)(y_1^2 + y_2^2 + y_3^2 + y_4^2) =$
$(x_1y_1 + x_2y_2 + x_3y_3 + x_4y_4)^2 + (x_1y_2 - x_2y_1 + x_3y_4 - x_4y_3)^2$
$+ (x_1y_3 - x_3y_1 + x_4y_2 - x_2y_4)^2 + (x_1y_4 - x_4y_1 + x_2y_3 - x_3y_2)^2$
It may be derived from the property of quaternions that the norm of the product is equal to the product of the norms.