  This paper is a continuation of our paper math.QA/0403010 at which several
coset subalgebras of the lattice VOA $V_{\sqrt{2}E_8}$ were constructed and the
relationship between such algebras with the famous McKay observation on the
extended E_8 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
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
VOA. The existence of U inside the Moonshine VOA for the cases of 1A, 2A, 2B
and 4A is also established. Moreover, the cases for 3A, 5A and 3C are
discussed.
