We are all familiar with Fibonacci-Brahmagupta’s identity:

$$(a^2-mb^2)(c^2-md^2)=(ac+ mbd)^2-m(ad+bc)^2$$

I am trying to find whether there is a similar identity:

$$(a^2-mb^2)(c^2-nd^2)=x^2-py^2$$ where $p,m,n$ are not all equal.

If this problem has already been solved, please save me the trouble by pointing me at a reference. Otherwise, any hint will be greatly appreciated.Thanks.