  Let $G$ be a simple complex Lie group with Lie algebra $\mf g$ and let $\af$
be the affine Lie algebra. We use intertwining operators and
Knizhnik-Zamolodchikov equations to construct a family of $\N$-graded vertex
operator algebras associated to $\mf g$. They are $\af \oplus \af$-modules of
dual levels $k, \bar k \notin \Q$ in the sense that $k + \bar k = -2 h^\vee$
where $h^\vee$ is the dual Coxeter number of $\mf g$. Its conformal weight 0
component is the algebra of regular functions on $G$. This family of vertex
operator algebras were previously studied by Arkhipov-Gaitsgory and
Gorbounov-Malikov-Schechtman from different points of view. We show that the
vertex envelope of the vertex algebroid associated to $G$ and level $k$ is
isomorphic to the vertex operator algebra we constructed above when $k$ is
irrational. The case of integral central charges is also discussed.
