These identities hold for all integers, as well as all rational numbers; more generally, they are true in any commutative ring.
Analogous identities are Euler's four-square related to quaternions, and Degen's eight-square derived from the octonions which has connections to Bott periodicity.
These identities are strongly related with Hurwitz's classification of composition algebras.
If a, b, c, and d are real numbers, the Brahmagupta–Fibonacci identity is equivalent to the multiplicative property for absolute values of complex numbers: This can be seen as follows: expanding the right side and squaring both sides, the multiplication property is equivalent to and by the definition of absolute value this is in turn equivalent to An equivalent calculation in the case that the variables a, b, c, and d are rational numbers shows the identity may be interpreted as the statement that the norm in the field Q(i) is multiplicative: the norm is given by and the multiplicativity calculation is the same as the preceding one.
In its original context, Brahmagupta applied his discovery of this identity to the solution of Pell's equation x2 − Ay2 = 1.