  We study McKay's observation on the Monster simple group, which relates the
2A-involutions of the Monster simple group to the extended E_8 diagram, using
the theory of vertex operator algebras (VOAs). We first consider the
sublattices L of the E_8 lattice obtained by removing one node from the
extended E_8 diagram at each time. We then construct a certain coset (or
commutant) subalgebra U associated with L in the lattice VOA V_{\sqrt{2}E_8}.
There are two natural conformal vectors of central charge 1/2 in U such that
their inner product is exactly the value predicted by Conway. The Griess
algebra of U coincides with the algebra described in Conway's paper. There is a
canonical automorphism of U of order |E_8/L|. Such an automorphism can be
extended to the Leech lattice VOA V_\Lambda and it is in fact a product of two
Miyamoto involutions. In the sequel [LYY] to this article we shall develop the
representation theory of $U$. It is expected that if U is actually contained in
the Moonshine VOA V^\natural, the product of two Miyamoto involutions is in the
desired conjugacy class of the Monster simple group.
