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.