Untwisted affine central extension

The untwisted central extension is the vector space $\overline{L\mathfrak{g}}=L\mathfrak{g}\oplus \mathbb{C}c$, with $c$ a new nonzero basis vector, and bracket $[u+ac,v+bc]=[u,v{]}_{L\mathfrak{g}}+\omega (u,v)c\mathrm{.}$ Here $L\mathfrak{g}$ is Loop algebra of a simple Lie algebra, while $B$ and $\omega$ are exactly the normalized invariant form and residue cocycle of Residue two cocycle on a loop algebra. Their alternatingness and cocycle identity are proved in The loop residue form is alternating and The loop residue form satisfies the Lie two cocycle identity. Thus $c$ is central, and for $x_m=x\otimes t^m$, $[x_m,y_n]=[x,y{]}_{m+n}+m{\delta }_{m,-n}B(x,y)c\mathrm{.}$ The bracket is bilinear and alternating by the first cited lemma. Its Jacobi sum has loop component zero by loop Jacobi and central component $\omega ([u,v],w)+\omega ([v,w],u)+\omega ([w,u],v)=0$ by the second lemma. Central inputs give zero directly. Hence this defines a Lie algebra. The projection to $L\mathfrak{g}$ is a surjective Lie homomorphism with kernel $\mathbb{C}c$. The vector-space inclusion of the loop algebra need not preserve its bracket. No universal property of this extension is asserted.