  Let $H$ be the quaternion algebra. Let $g$ be a complex Lie algebra and let
$U(g)$ be the enveloping algebra of $g$. We define a Lie algebra structure on
the tensor product space of $H$ and $U(g)$, and obtain the quaternification
$g^H$ of $g$. Let $S^3g^H$ be the set of $g^H$-valued smooth mappings over
$S^3$. The Lie algebra structure on $S^3g^H$ is induced naturally from that of
$g^H$. On $S^3$ exists the space of Laurent polynomial spinors spanned by a
complete orthogonal system of eigen spinors of the tangential Dirac operator on
$S^3$. Tensoring $U(g)$ we have the space of $U(g)$-valued Laurent polynomial
spinors, which is a Lie subalgebra of $S^3g^H$. We introduce a 2-cocycle on the
space of $U(g)$-valued Laurent polynomial spinors by the aid of a tangential
vector field on $S^3$. Then we have the corresponding central extension $\hat
g(a)$ of the Lie algebra of $U(g)$-valued Laurent polynomial spinors. Finally
we have the a Lie algebra $\hat g=\hat g(a)+Cd$ which is obtained by adding to
$\hat g(a)$ a derivation $d$ which acts on $\hat g(a)$ as the radial
derivation. When $g$ is a simple Lie algebra with its Cartan subalgebra $h$, We
shall investigate the weight space decomposition of $(\hat g, ad(\hat h))$,
where $\hat h=h+Ca+Cd$ . The previous versions (v1-v7) of this article
contained several incorrect assertions and here we have corrected them.
