Sums of squares, the octonions, and (7,3,1) - MAA Sections

Four squares
The identity, due to Euler (1748):
(a 2 1 + a 2 2 + a 2 3 + a 2 4)(b 2 1 + b 2 2 + b 2 3 + b 2 4) =
= (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
The algebra: the quaternions
H = {a 1 + a 2 i + a 3 j + a 4 k|a 1 , a 2 , a 3 , a 4 ∈ R, i 2 = j 2 = k 2 = ijk = −1}
The norm: N(a 1 + a 2 i + a 3 j + a 4 k) = a 2 1 + a2 2 + a2 3 + a2 4
The origins: W. R. Hamilton, who —
failed in an attempt to define a multiplication on R 3 , and then —
succeeded in defining a noncommutative multiplication on R 4 .
Brown
The Octonions