Candido's identity

Candido's identity, named after the Italian mathematician Giacomo Candido, is an identity for real numbers. It states that for two arbitrary real numbers ${\displaystyle x}$ and ${\displaystyle y}$ the following equality holds:^[2]

${\displaystyle \left[x^{2}+y^{2}+(x+y)^{2}\right]^{2}=2[x^{4}+y^{4}+(x+y)^{4}]}$

The identity however is not restricted to real numbers but holds in every commutative ring.^[2]

Candido originally devised the identity to prove the following identity for Fibonacci numbers:^[1]

${\displaystyle (f_{n}^{2}+f_{n+1}^{2}+f_{n+2}^{2})^{2}=2(f_{n}^{4}+f_{n+1}^{4}+f_{n+2}^{4})}$