On the ideal $J[\kappa ]$

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 ${\mathrm{\aleph }}_{n+1}$-Souslin tree.