Euler four-square identity

The 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)=$${(x_1{y}_1+x_2{y}_2+x_3{y}_3+x_4{y}_4)}^2+{(x_1{y}_2-x_2{y}_1+x_3{y}_4-x_4{y}_3)}^2$$+{(x_1{y}_3-x_3{y}_1+x_4{y}_2-x_2{y}_4)}^2+{(x_1{y}_4-x_4{y}_1+x_2{y}_3-x_3{y}_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.