Euler's Four-Square Identity
The Four-Square Identity, credited to Leonhard Euler, states that for any eight numbers , we must have (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.
Define and . Recall that the quaternion norm of a number , written as , is simply .
Then Euler's Four-Square Identity simply reads ; i.e. the quaternion norm is multiplicative.