Vertex operator algebra analogue of embedding of B4B4 into F4F4

Let LB(−52,0) (resp. LF(−52,0)) be the simple vertex operator algebra associated with affine Lie algebra of type B4(1) (resp. F4(1)) with the lowest admissible half-integer level −52. We show that LB(−52,0) is a vertex subalgebra of LF(−52,0) with the same conformal vector, and that LF(−52,0) is isomorphic to the extension of LB(−52,0) by its only irreducible module other than itself. We also study the representation theory of LF(−52,0), and determine the decompositions of irreducible weak LF(−52,0)-modules from the category OO into direct sums of irreducible weak LB(−52,0)-modules.