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A parametrization of solutions is

$x=2ad$

$y=2bd$

$z=2cd$

$t=a^2+b^2+c^2-d^2$

$w=a^2+b^2+c^2+d^2\ .$

It is easy to see that this generates all rational solutions if $a, b, c, d$ are rational numbers, and (consequently) all integers solutions up to a similarity factor, if $a, b, c, d$ are integers.

 for what it's worth, here's something more symmetric (with the same rmk as before)

$x=-a^2+b^2+c^2+d^2-2a(b+c+d)$

$y=\phantom{-}a^2-b^2+c^2+d^2-2b(a+c+d)$

$z=\phantom{-}a^2+b^2-c^2+d^2-2c(a+b+d)$

$t=\phantom{-}a^2+b^2+c^2-d^2-2d(a+b+c)$

$w=\phantom{-}2(a^2+b^2+c^2+d^2)\ .$

1

A parametrization of solutions is

$x=2ad$

$y=2bd$

$z=2cd$

$t=a^2+b^2+c^2-d^2$

$w=a^2+b^2+c^2+d^2\ .$

It is easy to see that this generates all rational solutions if $a, b, c, d$ are rational numbers, and (consequently) all integers solutions up to a similarity factor, if $a, b, c, d$ are integers.