Provenance of a result on regular simplices with integer vertices

Question

There are several MO questions related to the question of characterizing those integers $n$ for which there exists a regular $n$-simplex in $\mathbb{R}^n$ with integer vertices, e.g., coordinates of vertices of regular simplex and For which $n$ is there a regular $n$-simplex with vertices in $\mathbb{Z}^n$?. I would like to know where this result originally appeared.

Answer 1

I can answer my own question. The reference is I. J. Schoenberg, Regular simplices and quadratic forms, J. London Math. Soc. 12 (1937), 48-55.