Embeddings of vertex operator algebras associated to orthogonal affine Lie algebras

Let $L_{D_{\mathrm{\ell }}}(-\mathrm{\ell }+3\mathrm{/}2,0)$ (resp. $L_{B_{\mathrm{\ell }}}(-\mathrm{\ell }+3\mathrm{/}2,0)$) be the simple vertex operator algebra associated to affine Lie algebra of type $D_{\mathrm{\ell }}^{(1)}$ (resp. $B_{\mathrm{\ell }}^{(1)}$) with the lowest admissible half-integer level $-\mathrm{\ell }+3\mathrm{/}2$. We show that $L_{D_{\mathrm{\ell }}}(-\mathrm{\ell }+3\mathrm{/}2,0)$ is a vertex subalgebra of $L_{B_{\mathrm{\ell }}}(-\mathrm{\ell }+3\mathrm{/}2,0)$ with the same conformal vector. For $\mathrm{\ell }=4$, $L_{D_4}(-5\mathrm{/}2,0)$ is a vertex subalgebra of three copies of $L_{B_4}(-5\mathrm{/}2,0)$ contained in $L_{F_4}(-5\mathrm{/}2,0)$, and all five of these vertex operator algebras have the same conformal vector.