Is it possible to generalise Brahmagupta's identity
further, by which I mean, for different n, for example take the equations:
$$1357 = 37^2 - 3\times2^2$$
$$1357 = 38^2 - 87\times1^2$$
$$1357^2 = 1841449 = 3445^2 - 261\times196^2$$
Is there an algebraic route between the first two and the third?
No, I wouldn't think so. It would be easy enough if $3 \cdot 87$ were a square, this is called composition in different orders in Buell's book.
As $261$ is not a square: if there were some nice way to combine things, we could expect to take
$$ 2^2 - 3 \cdot 1^2 = 1 \; , \; \; \; 28^2 - 87 \cdot 3^2 = 1 $$
and quickly solve $x^2 - 261 y^2 = 1^2 = 1$ in fairly small numbers. However, the smallest such is
$$ 192119201^2 - 261 \cdot 11891880^2 = 1 $$