Structure of affine Lie algebras It is well-known that to every simple Lie algebra $\mathfrak{g}$ one can associate an affine Kac-Moody algebra by a double extension (once by a 2-cocycle and once by a derivation). One can then show that this algebra is the Kac-Moody algebra associated to the extended Cartan matrix of $\mathfrak{g}$.
However it is the intermediate Lie algebra, obtained by the first central extension, that is of interest to me. This Lie algebra $L\mathfrak{g}\oplus\mathbb{C}c$ (where $L\mathfrak{g}$ is the loop algebra of $\mathfrak{g}$) with bracket $$[g\otimes t^k+\lambda c, g'\otimes t^l+\mu c] = [g\otimes t^k, g'\otimes t^l]_{L\mathfrak{g}} + k(g|g')\delta_{k+l, 0}$$is often called the ''affine Lie algebra'' associated to $\mathfrak{g}$.
Since in some papers the further extension by a derivation is omitted I was wondering what the structure of this ''affine Lie algebra'' is. Does this algebra admit a generalized Cartan matrix (is it a Kac-Moody algebra on its own)? And perhaps a secondary question, why do some authors (especially in the physics literature) omit this second extension?
 A: First, you should know why we need such extension by a (degree) derivation sometimes. In another word, why the dimension of Cartan subalgebra $\mathfrak{h}$ is $2n-\mathrm{rank}(A)$, where $A$ is a $n \times n$-GCM. In the affine case, $\mathrm{rank}(A)=n-1$, that is, the dimension of $\mathfrak{h}$ is $n+1$. I think this topic realization of Kac-Moody algebras in MO may give you some answers.
The "affine Lie algebra" mentioned in your question is just the derived subalgebra
$$\mathfrak{g}'(A)=[\mathfrak{g}(A),\mathfrak{g}(A)],$$
where $\mathfrak{g}(A)$ is the normal affine Kac-Moody Lie algebra. Actually, the derived subalgebra can be generated by the Chevalley generators $\{e_i, f_i, i=1,2,\cdots, n\}$ of $\mathfrak{g}(A)$.
Second, the structure of $\mathfrak{g}'(A)$. Let $A=(a_{ij})$ be a symmetrizable GCM, then it is well-known that any derived subalgebra $\mathfrak{g}'(A)$ is the quotient of the free Lie algebra with generators $\{e_i, f_i, h_i, i=1,2, \cdots, n \}$ and the following defining relations:


*

*$[e_i, f_j]= \delta_{ij}h_i$,

*$[h_i, h_j]=0$,

*$[h_i, e_j] = a_{ij}e_j$,

*$[h_i,f_j] = -a_{ij}f_j$,

*Serre's relations.


Since each affine GCM is symmetrizable,  $\mathfrak{g}'(A)$ associated to an affine GCM can be described like this. In addition, we may think that the affine Lie algebra is uniquely determined by the affine GCM (up to degree derivations).
