Could you state or direct me to results regarding the Diophantine equation $a^2+b^2=c^2+d^2$ over integers? Specifically, I am looking for a complete parametrization. In the case that a complete parametrization does not exist, I would be interested in seeing complete parametrizations for special cases.
For example, in the case that any one of the variables is zero, we are led to Pythagorean triples. Taking a variable to be some non-zero constant would be interesting. I'm looking for known results as I am not as familiar with the literature as many MO-ers out there.