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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?

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