integer quadruples with all pairwise distances being squares

[Max Alekseyev]

SeqFaq,

There are quite interesting recent findings of quadruples of distinct
integers with all six pairwise distances being squares:
http://www.mathlinks.ro/viewtopic.php?t=33650

It is clear that the smallest element of such a quadruple can be taken
equal 0 (by shifting all 4 elements). Then the other elements are
perfect squares themselves and so are their pairwise distances.
In other words, each such quadruple corresponds to an unique triple of
distinct squares (x^2,y^2,z^2) such that x^2<y^2<z^2 and each of
y^2-x^2, z^2-x^2, z^2-y^2 is also a square.

Would anybody like to compute such triples (say, ordered by the value
of z) and add them to OEIS?

Thanks,
Max