Brahmagupta's identity

In algebra, Brahmagupta's identity says that, for given n, the product of two numbers of the form a² + nb² is itself a number of that form. In other words, the set of such numbers is closed under multiplication. Specifically:

$${\displaystyle (a^{2}+nb^{2})(c^{2}+nd^{2})=(ac\pm nbd)^{2}+n(ad\mp bc)^{2}.}$$

This identity holds in both the ring of integers and the ring of rational numbers, and more generally in any commutative ring.

Also, it can be viewed as a special case of the following identity:

$${\displaystyle (ha^{2}+kb^{2})(hc^{2}+kd^{2})=(hac\pm kbd)^{2}+hk(ad\mp bc)^{2}.}$$