  In [FFR] Feingold, Frenkel and Ries gave a spinor construction of the vertex
operator para-algebra (abelian intertwining algebra) V = V^0 \oplus V^1 \oplus
V^2 \oplus V^3, whose summands are four level-1 irreducible representations of
the affine Kac-Moody algebra D_4^(1). The triality group S_3 = < \sigma,\tau |
\sigma^3 = 1 = \tau^2, \tau\sigma\tau = \sigma^{-1} > in Aut(V) was
constructed, preserving V^0 and permuting the V^i, for i=1,2,3. V is
(1/2)Z-graded where V^i_n denotes the n-graded subspace of V^i. Vertex
operators Y(v,z) for v in V^0_1 represent D_4^(1) on V, while those for which
\sigma(v) = v represent G_2^(1).
  We investigate branching rules, how V decomposes into a direct sum of
irreducible G_2^(1) representations. We use a two-step process, first
decomposing with respect to the intermediate subalgebra B_3^(1), represented by
Y(v,z) for \tau(v) = v. There are three vertex operators, Y(\omega_{D_4},z),
Y(\omega_{B_3},z), and Y(\omega_{G_2},z) each representing the Virasoro algebra
given by the Sugawara constructions from the three algebras. The
Goddard-Kent-Olive coset construction [GKO] gives two mutually commuting coset
Virasoro representations, provided by the vertex operators
Y(\omega_{D_4}-\omega_{B_3},z) and Y(\omega_{B_3}-\omega_{G_2},z), with central
charges 1/2 and 7/10, respectively. The first one commutes with B_3^(1), and
the second one commutes with G_2^(1). This gives the space of highest weight
vectors for G_2^(1) in V as tensor products of irreducible Virasoro modules
L(1/2,h^{1/2}) \otimes L(7/10,h^{7/10}). This dissertation explicitly
constructs these coset Virasoro operators, and uses them to describe the
decomposition of V with respect to G_2^(1). This work also provides spinor
constructions of the 7/10 Virasoro modules, and of the two level-1
representations of G_2^(1) inside V.
