A note on representations of some affine vertex algebras of type D

In this note we construct a series of singular vectors in universal affine vertex operator algebras associated to $D_{\mathrm{\ell }}^{(1)}$ of levels $n-\mathrm{\ell }+1$, for $n\in {\mathbb{Z}}_{>0}$. For $n=1$, we study the representation theory of the quotient vertex operator algebra modulo the ideal generated by that singular vector. In the case $\mathrm{\ell }=4$, we show that the adjoint module is the unique irreducible ordinary module for simple vertex operator algebra $L_{D_4}(-2,0)$. We also show that the maximal ideal in associated universal affine vertex algebra is generated by three singular vectors.