Is it possible to find an infinite set of points in the 4D, 5D... nD space where the distance between any pair is rational?

Question

This question is a generalisation of the 3D space question asked a few days ago: 3D SPACE - RATIONAL DISTANCE
Of course, the points must not belong to the same hyperplane.

Answer 1

The thread already gives you he answer. Just construct the infinite family in the unitary circle of the plane (say, using the first coordinates), then join the points of the form $\sqrt{8} e_j $ for $j \geq 3$. Interesting generalization though, it's nice to know that there is such set out there.