integer solutions to $x^2+y^2+z^2+t^2 = w^2$ Is there a way to find all integer primitive solutions to the equation $x^2+y^2+z^2+t^2 = w^2$? i.e., is there a parametrization which covers all the possible solutions?

For the equation:
$X^2+Y^2+Z^2+Q^2=R^2$
We can write the solution:
$X=2p^2+2(a+b+t)ps+(ab+at+bt)s^2$
$Y=2p^2+2(b+t-a)ps+(bt-a^2)s^2$
$Z=2p^2+2(a+t-b)ps+(at-b^2)s^2$
$Q=2p^2+2(a+b-t)ps+(ab-t^2)s^2$
$R=4p^2+2(a+b+t)ps+(a^2+b^2+t^2)s^2$
And more:
$X=-2p^2+2(a+b+t)ps+(4a^2+4b^2+4t^2-ab-at-bt)s^2$
$Y=-2p^2+2(b+t-5a)ps+(a^2-4a(b+t)+4b^2+4t^2-bt)s^2$
$Z=-2p^2+2(t+a-5b)ps+(b^2-4b(a+t)+4a^2+4t^2-at)s^2$
$Q=-2p^2+2(a+b-5t)ps+(t^2-4t(a+b)+4a^2+4b^2-ab)s^2$
$R=4p^2+2(a+b+t)ps+(7a^2+7b^2+7t^2-4ab-4at-4bt)s^2$
$a,b,t,p,s$ - integers asked us.
