Mckay's observation and vertex operator algebras generated by two conformal vectors of central charge 1/2

This paper is a continuation of (33) at which several coset subalgebras of the lattice VOA Vp 2E8 were constructed and the relationship between such algebras with the famous McKay observation on the extended E8 diagram and the Monster simple group were discussed. In this article, we shall provide the technical details. We completely determine the structure of the coset subalgebras constructed and show that they are all generated by two conformal vectors of central charge 1/2. We also study the represen- tation theory of these coset subalgebras and show that the product of two Miyamoto involutions is in the desired conjugacy class of the Monster simple group if a coset subal- gebra U is actually contained in the Moonshine VOA V ♮ . The existence of U inside the Moonshine VOA V ♮ for the cases of 1A,2A,2B and 4A is also established. Moreover, the cases for 3A, 5A and 3C are discussed. and the Monster simple group were discussed. In this article, we shall provide the technical details. We shall determine the structure of the coset subalgebras and show that they are all generated by two conformal vectors of central charge 1/2. We also study the representation theory of these coset subalgebras and show that the product of two Miyamoto involutions is in the desired conjugacy class of the Monster simple group if a coset subalgebra U is actually contained in the Moonshine