  Let $R =k[x_1, \cdots, x_n,\cdots]$ be the infinite variable polynomial ring
equipped with the natural $\mathfrak{S}_\infty$ action, where $k$ is a field of
characteristic zero. In recent work \cite{NS21}, Nagpal--Snowden gave an
indirect proof that $\mathfrak{S}_\infty$-ideal generated by $(x_1-x_2)^{2n}$
is not $\mathfrak{S}_\infty$-prime. In this paper, we give a direct proof, with
explicit elements. We further formulate some conjectures on possible
generalizations of the result.
