Vertex operator algebra analogue of embedding of B4 into F4

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