Euler's Four-Square Identity

Identity

The Four-Square Identity, credited to Leonhard Euler, states that for any eight 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 statement can be easily verified by expansion.) In other words, the product of the sums of four squares is itself the sum of four squares.

Quaternionic interpretation

Define $X := x_1 + x_2 i + x_3 j + x_4 k$ and $Y: = y_1 + y_2 i + y_3 j + y_4 k$. Recall that the quaternion norm of a number $a + bi + cj + dk$, written as $|a + bi + cj + dk|^2$, is simply $a^2 + b^2 + c^2 + d^2$.

Then Euler's Four-Square Identity simply reads $|XY|^2 = |X|^2 |Y|^2$; i.e. the quaternion norm is multiplicative.

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