  Motivated by a question from a recent paper by Gilton, Levine and
Stejskalova, we obtain a new characterization of the ideal $J[\kappa]$, from
which we confirm that $\kappa$-Souslin trees exist in various models of
interest. As a corollary we get that for every integer $n$ such that $\mathfrak
b<2^{\aleph_n}=\aleph_{n+1}$, if $\square(\aleph_{n+1})$ holds, then there
exists an $\aleph_{n+1}$-Souslin tree.
