The identity of eight squares is the following identity expressing the product of the sums of eight squares as the sum of eight squares:
{\ displaystyle {\ begin {aligned} (a_ {1} ^ {2} + a_ {2} ^ {2} + a_ {3} ^ {2} + a_ {4} ^ {2} + a_ {5} ^ {2} + a_ {6} ^ {2} + a_ {7} ^ {2} + a_ {8} ^ {2}) & \ cdot (b_ {1} ^ {2} + b_ {2} ^ {2} + b_ {3} ^ {2} + b_ {4} ^ {2} + b_ {5} ^ {2} + b_ {6} ^ {2} + b_ {7} ^ {2} + b_ {8} ^ {2}) = \\ & = (a_ {1} b_ {1} -a_ {2} b_ {2} -a_ {3} b_ {3} -a_ {4} b_ {4} - a_ {5} b_ {5} -a_ {6} b_ {6} -a_ {7} b_ {7} -a_ {8} b_ {8}) ^ {2} + \\ & + (a_ {2} b_ {1} + a_ {1} b_ {2} + a_ {4} b_ {3} -a_ {3} b_ {4} + a_ {6} b_ {5} -a_ {5} b_ {6} - a_ {8} b_ {7} + a_ {7} b_ {8}) ^ {2} + \\ & + (a_ {3} b_ {1} -a_ {4} b_ {2} + a_ {1} b_ {3} + a_ {2} b_ {4} + a_ {7} b_ {5} + a_ {8} b_ {6} -a_ {5} b_ {7} -a_ {6} b_ {8}) ^ {2} + \\ & + (a_ {4} b_ {1} + a_ {3} b_ {2} -a_ {2} b_ {3} + a_ {1} b_ {4} + a_ {8} b_ {5} -a_ {7} b_ {6} + a_ {6} b_ {7} -a_ {5} b_ {8}) ^ {2} + \\ & + (a_ {5} b_ {1} -a_ {6} b_ {2} -a_ {7} b_ {3} -a_ {8} b_ {4} + a_ {1} b_ {5} + a_ {2} b_ {6} + a_ {3} b_ {7} + a_ {4} b_ {8}) ^ {2} + \\ & + (a_ {6} b_ {1} + a_ {5} b_ {2} -a_ {8} b_ {3} + a_ {7} b_ {4} -a_ {2} b_ {5} + a_ {1} b_ {6} -a_ {4} b_ {7} + a_ {3} b_ {8}) ^ {2} + \\ & + (a_ {7} b_ {1} + a_ {8} b_ {2} + a_ {5} b_ {3} -a_ {6} b_ {4} -a_ {3} b_ {5} + a_ {4} b_ {6} + a_ {1} b_ {7} -a_ {2} b_ {8}) ^ {2} + \\ & + (a_ {8} b_ {1} -a_ {7 } b_ {2} + a_ {6} b_ {3} + a_ {5} b_ {4} -a_ {4} b_ {5} -a_ {3} b_ {6} + a_ {2} b_ {7} + a_ {1} b_ {8}) ^ {2}. \ end {aligned}}} 
First discovered by the Danish mathematician Ferdinand Degen around 1818 , this remarkable identity was “rediscovered” two more times: first by Thomas Graves in 1843 , and then by Arthur Cayley in 1845 . Cayley derived it by working on a generalization of quaternions called octonions . In algebraic terms, identity means that the norm of the product of two octonions is equal to the product of their norms: {\ displaystyle \ | a \ cdot b | | = \ | a \ | \ cdot \ | b \ |} 
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A similar statement is true for quaternions ( “the identity of four squares” ), complex numbers ( “the identity of two squares” ) and real numbers. In 1898, Adolf Hurwitz proved that such an identity does not exist either for 16 ( sedenions ) or for any other number of squares except 1, 2, 4, and 8.