  In this article, we study the moonshine vertex operator algebra starting with
the tensor product of three copies of the vertex operator algebra
$V_{\sqrt2E_8}^+$, and describe it by the quadratic space over $\F_2$
associated to $V_{\sqrt2E_8}^+$. Using quadratic spaces and orthogonal groups,
we show the transitivity of the automorphism group of the moonshine vertex
operator algebra on the set of all full vertex operator subalgebras isomorphic
to the tensor product of three copies of $V_{\sqrt2E_8}^+$, and determine the
stabilizer of such a vertex operator subalgebra. Our approach is a vertex
operator algebra analogue of "An $E_8$-approach to the Leech lattice and the
Conway group" by Lepowsky and Meurman. Moreover, we find new analogies among
the moonshine vertex operator algebra, the Leech lattice and the extended
binary Golay code.
