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

Eight squares, continued
The algebra: the octonions O, where
O = {a 0 + a 1 e 1 + . . . + a 7 e 7 |a 0 , . . . , a 7 ∈ R, e 2 t = −1}; the e t are
called the octonion units
The norm:
N(a 0 + a 1 e 1 + · · · + a 7 e 7 ) = a 2 1 + a2 1 + a2 2 + a2 3 + a2 4 + a2 5 + a2 6 + a2 7
The origins: J. T. Graves, who —
received Hamilton’s letter announcing the quaternions, and then —
went him one better by defining a noncommutative and nonassociative
multiplication on R 8 .
The multiplication is given by the following table.
Brown
The Octonions