Citation for isometric embeddability of $\ell_2$ into $\ell_p^\binom{n}{2}$ for $p \geq 1$?
This paper by Keith Ball seems to be what you are looking for:


  Ball, Keith. "Isometric embedding in $\ell_p$-spaces." 
  European Journal of Combinatorics 11.4 (1990): 305-311.


Link to the paper here:
https://www.sciencedirect.com/science/article/pii/S019566981380131X