Vertex operator algebra analogue of embedding of $B_4$ into $F_4$

Let L_{B}(-5/2,0) (resp. L_{F}(-5/2,0)) be the simple vertex operator algebra associated to affine Lie algebra of type $B_{4}^{(1)}$ (resp. $F_{4}^{(1)}$) with the lowest admissible half-integer level -5/2. We show that L_{B}(-5/2,0) is a vertex subalgebra of L_{F}(-5/2,0) with the same conformal vector, and that L_{F}(-5/2,0) is isomorphic to the extension of L_{B}(-5/2,0) by its only irreducible module other than itself. We also study the representation theory of L_{F}(-5/2,0), and determine the decompositions of irreducible weak L_{F}(-5/2,0)-modules from the category $\mathcal{O}$ into direct sums of irreducible weak L_{B}(-5/2,0)-modules.