Degen's eight-square identity

Degen's eight-square identity expresses the product of two sums of eight squares as a sum of eight squares. It follows from the multiplicativity of the octonion norm.

Connect

Octonion: An eight-dimensional real division algebra whose multiplication is nonassociative but alternative. Its norm multiplicativity yields the eight-square product formula.
Ferdinand Georg Frobenius: A German mathematician known for foundational work in algebra and representation theory. His 1878 theorem classified finite-dimensional associative real division algebras, framing the algebraic context.
Sum of squares: An expression formed by adding squares of numbers or algebraic quantities. Each side of Degen's identity is organized as a sum of eight such terms.
Sum of squares theorem: A theorem or result describing which numbers or forms can be represented as sums of squares. Degen's formula gives a multiplication rule for representations by eight squares.
Octonion norm: The Euclidean norm on octonions, whose square is the sum of the squares of eight real coordinates. Multiplying octonions preserves the product of their norms, giving the identity.
Leonhard Euler: An eighteenth-century Swiss mathematician whose work ranged across analysis, number theory, and mechanics. His four-square identity is a prominent predecessor in the history of product formulas.
Multiplicative norm: A norm satisfying that the norm of a product equals the product of the norms. This property converts octonion multiplication directly into the eight-square equation.
Composition of quadratic forms: A polynomial identity showing that products of values of quadratic forms are represented by another quadratic form. The identity is a classical composition law for the eight-variable sum-of-squares form.
Composition algebra: An algebra equipped with a quadratic form satisfying a multiplicative norm law. Degen's identity is the coordinate form of a composition algebra norm law.
Adolf Hurwitz: A German mathematician who proved a classification theorem for quadratic forms with multiplicative composition. His work placed eight-square multiplication within the theory of composition.