Brahmagupta–Fibonacci identity

Statement

Let $a$, $b$, $c$ and $d$ be elements of a commutative ring. Then $$(a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2 = (ac + bd)^2 + (ad - bc)^2.$$

Remarks

So the product of two sums of two squares is again a sum of two squares. For example, $5 \cdot 13 = 65$, and the identity gives $65 = 1^2 + 8^2 = 7^2 + 4^2$. For complex numbers, the identity says $|z|^2 |w|^2 = |zw|^2$ with $z = a + bi$ and $w = c + di$. It is a step in the proof of the sum of two squares theorem.

History

The identity appears in the Arithmetica of Diophantus, of the third century. Brahmagupta found it again in the seventh century, and Fibonacci published it in Europe in 1225.

Explanations

L1 human

If two numbers are sums of two squares, then so is their product. You can check the identity by expanding both sides: the cross terms $2abcd$ cancel.