The Hurwitz Theorem on Sums of Squares by Linear Algebra

(1.1) (x1 + x 2 2)(y 2 1 + y 2 2) = (x1y1 − x2y2) + (x1y2 + x2y1). There is also an identity like this for a sum of four squares: (x1 + x 2 2 + x 2 3 + x 2 4)(y 2 1 + y 2 2 + y 2 3 + y 2 4) = (x1y1 − x2y2 − x3y3 − x4y4) + (1.2) (x1y2 + x2y1 + x3y4 − x4y3) + (x1y3 + x3y1 − x2y4 + x4y2) + (x1y4 + x4y1 + x2y3 − x3y2). This was discovered by Euler in the 18th century, forgotten, and then rediscovered in the 19th century by Hamilton in his work on quaternions. Shortly after Hamilton’s rediscovery of (1.2) Cayley discovered a similar 8-square identity. In all of these sum-of-squares identities, the terms being squared on the right side are all bilinear expressions in the x’s and y’s: each such expression, like x1y2 +x2y1 for sums of two squares, is a linear combination of the x’s when the y’s are fixed and a linear combination of the y’s when the x’s are fixed. It was natural for mathematicians to search for a similar 16-square identity next, but they were unsuccessful. At the end of the 19th century Hurwitz [3] proved his famous “1,2,4,8 theorem,” which says that further identities of this kind are impossible.