New solutions to the Hurwitz problem on square identities

The Hurwitz problem of composition of quadratic forms, or of “sum of squares identity” is tackled with the help of a particular class of (Z/2Z)n-graded non-associative algebras generalizing the octonions. This method provides explicit formulas for the classical Hurwitz–Radon identities and leads to new solutions in a neighborhood of the Hurwitz–Radon identities.