Degree derivation and full untwisted affine algebra

On Untwisted affine central extension define $D(x_m)=m{x}_m$ and $D(c)=0$, extending linearly. For two modes the loop component of $D[x_m,y_n]$ is $(m+n)[x,y{]}_{m+n}$, the same as that of $[D{x}_m,y_n]+[x_m,D{y}_n]$. The latter's central coefficient is $(m+n)m{\delta }_{m,-n}B(x,y)=0$, while $D$ kills the former's central term. Thus $D$ is a derivation; the cases involving $c$ vanish directly.

The full untwisted affine algebra is $\widehat{\mathfrak{g}}=\overline{L\mathfrak{g}}\oplus \mathbb{C}d,$ where $d$ is a new basis vector, with bracket $[u+ad,v+bd]=[u,v]+aD(v)-bD(u)\mathrm{.}$ In particular $[d,x_m]=m{x}_m$, $[d,c]=0$ and $[d,d]=0$. Jacobi with no $d$ is the central-extension identity; with one $d$ it is precisely the derivation identity just checked; with two $d$ the terms $D^2(u)$ cancel; with three $d$ it is zero. Multilinearity proves Jacobi generally. This is an algebraic semidirect extension; both $u$ and $v$ have finite Laurent support. No AC is needed.