Vertex operator algebra analogue of embedding of B₄ into F₄

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Let LB(-5/2,0) (resp. LF(-5/2,0)) be the simple vertex operator algebra associated to affine Lie algebra of type B₄⁽¹⁾ (resp. F₄⁽¹⁾) with the lowest admissible half-integer level -5/2. We show that LB(-5/2,0) is a vertex subalgebra of LF(-5/2,0) with the same conformal vector, and that LF(-5/2,0) is isomorphic to the extension of LB(-5/2,0) by its only irreducible module other than itself. We also study the representation theory of LF(-5/2,0), and determine the decompositions of irreducible weak LF(-5/2,0)-modules from the category O into direct sums of irreducible weak LB(-5/2,0)-modules.