Loop and affine GCM presentations are isomorphic

F1 gives $[e_0,f_0]=h_0$ in the target. F2 gives the Cartan action with weights ${\alpha }_0=\delta -\theta$ and $-{\alpha }_0$, as well as commutativity of the Cartan. The finite simple brackets hold by F5. For $i>0$, $[e_0,f_i]$ lies at finite weight $-\theta -{\alpha }_i$ and $[e_i,f_0]$ at $\theta +{\alpha }_i$, both absent by highest-root maximality. Their mode degrees are nonzero, so no central term occurs. Thus every mixed relation $[e_i,f_j]={\delta }_{ij}{h}_i$ holds.

The finite positive Serre relations follow from finite root strings. For $i>0$, $f_{\theta }$ is a lowest vector for the $i$th finite triple, of weight $-\theta (h_i)={\widehat{a}}_{i0}$, because $-\theta -{\alpha }_i$ is absent. Its raising string is killed after $1-{\widehat{a}}_{i0}$ applications of $\mathrm{ad}{e}_i$. Conversely $e_i$ is a highest vector for the $\theta$ triple, of weight ${\alpha }_i({\theta }^{\vee })=-{\widehat{a}}_{0i}$, since $\theta +{\alpha }_i$ is absent. Its lowering string is killed after $1-{\widehat{a}}_{0i}$ applications of $\mathrm{ad}{f}_{\theta }$. If $\theta ={\alpha }_i$, this is the adjoint rank-one string $e_{\theta },h_{\theta },f_{\theta },0$ of length three. In the loop brackets the relevant positive mode degrees never produce a central term, so these are exactly the two Serre relations involving index zero. Interchanging raising and lowering and replacing every mode degree by its negative proves the negative Serre family by the same strings.

The matrix is symmetrizable by F1. Steps 1.1–1.2 and F3 therefore give a unique homomorphism $\phi :\mathfrak{g}(\widehat{A})\to \widehat{\mathfrak{g}}$ with the specified images. It fixes the embedded Cartan, so its kernel meets that Cartan trivially. F4 forces the kernel to be zero.

Its image contains $\mathfrak{g}\otimes 1$ by F5. The set $J_{+}=\{x\in \mathfrak{g}:x\otimes t\in \mathrm{im}\phi \}$ is an ideal in $\mathfrak{g}$, since $[y\otimes 1,x\otimes t]=[y,x]\otimes t$. It contains the nonzero $f_{\theta }$, hence is all of $\mathfrak{g}$ by simplicity. Likewise $J_{-}=\{x:x\otimes t^{-1}\in \mathrm{im}\phi \}$ contains $e_{\theta }$ and is all of $\mathfrak{g}$.

Nonabelian simplicity gives $[\mathfrak{g},\mathfrak{g}]=\mathfrak{g}$, because the derived algebra is a nonzero ideal. If all positive modes of degree $k-1$ lie in the image, then $[x\otimes t,y\otimes t^{k-1}]=[x,y]\otimes t^k$ for $k\ge 2$; finite sums of these brackets span the degree-$k$ mode. Induction from step 3.1 gives all positive modes. Bracketing degree $-1$ with degree $-(k-1)$ gives all negative modes in the same way. The image already contains $c,d$ through the Cartan. Hence it is the full affine algebra. Combined with step 2.1 this proves the isomorphism. All sums, string calculations and selections at a fixed mode are finite, with no AC.