The Lebesgue identity is the algebraic identity (a^2 + b^2 + c^2 + d^2)^2 = (a^2 + b^2 - c^2 - d^2)^2 + (2a c + 2b d)^2 + (2a d - 2b c)^2 (Nagell 1951, pp. 194-195).

Diophantine equation--2nd powers | Euler four-square identity