Omitting the central term breaks the affine GCM bracket

Take $e_0=f_{\theta }\otimes t$ and $f_0=e_{\theta }\otimes t^{-1}$. F1 gives $[e_0,f_0]=[f_{\theta },e_{\theta }]\otimes 1=-{\theta }^{\vee }$. If we adjoin $c$ as an independent central vector but leave this bracket unchanged, it still has zero $c$ coordinate. F2 instead requires $[e_0,f_0]=c-{\theta }^{\vee }$, whose $c$ coordinate is one. These vectors differ by the nonzero $c$.

Thus the required mixed relation fails and F3's realization cannot persist. In the unextended loop algebra there is not even a vector for this independent central coordinate. Sending $c$ to zero does produce a quotient representation of the derived affine algebra, but it cannot be the claimed faithful full realization. For finite ${\mathfrak{s}\mathfrak{l}}_2$ the same discrepancy is $-h$ versus $c-h$, already with degrees $1,-1$ and form value one. This explicit witness refutes the assertion without any choice assumption.