Euler's four-square identity

Statement

As an identity

In any commutative unital ring, if $a_{1},a_{2},a_{3},a_{4},b_{1},b_{2},b_{3},b_{4}$ are elements, then:

$(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2})(b_{1}^{2}+b_{2}^{2}+b_{3}^{2}+b_{4}^{2})=\,$

$(a_{1}b_{1}-a_{2}b_{2}-a_{3}b_{3}-a_{4}b_{4})^{2}+\,$

$(a_{1}b_{2}+a_{2}b_{1}+a_{3}b_{4}-a_{4}b_{3})^{2}+\,$

$(a_{1}b_{3}-a_{2}b_{4}+a_{3}b_{1}+a_{4}b_{2})^{2}+\,$

$(a_{1}b_{4}+a_{2}b_{3}-a_{3}b_{2}+a_{4}b_{1})^{2}.\,$

In terms of products two at a time

In any commutative unital ring, if $x$ and $y$ are both elements that can be expressed as a sum of four squares, then $xy$ can also be expressed as a sum of four squares.

In terms of products of arbitrary length

In any commutative unital ring, if $x_{1},x_{2},\dots ,x_{n}$ are all elements that can be expressed as a sum of four squares, then $x_{1}x_{2}\dots x_{n}$ can also be expressed as a sum of four squares.

Related facts

Similar facts

• Two-square identity
• Degen's eight-square identity

Applications

• Lagrange's four-square theorem