Odd-rank maximal ideals at collapsing levels of type $D$

We determine the defining ideal of the simple affine vertex algebra $L_{2-\mathrm{\ell }}({\mathfrak{s}\mathfrak{o}}_{2\mathrm{\ell }})$ for every odd $\mathrm{\ell }\ge 5$. Perše’s quadratic singular vector alone generates the maximal ideal of the universal affine vertex algebra at this level. Together with the established even-rank presentation, this gives a complete parity-dependent description of this type-$D$ collapsing family at $k=2-\mathrm{\ell }$: one quadratic generator in odd rank, and a quadratic generator together with two Pfaffian generators in even rank. The proof establishes a rank reduction for the quadratic quotients under minimal Drinfeld–Sokolov reduction, valid in both parities. Nonvanishing of reduction on nonzero graded subquotients then lifts simplicity along the odd-rank chain from the known base case $D_3\cong A_3$ at level $-1$.