I am following the notes (https://arxiv.org/abs/hep-th/9904145) to learn conformal field theory, and want to know how to derive the contributions to the Virasoro and Kac-Moody algebras from the central terms.
In their exposition, around Eq(3.25), the Kac-Moody algebra $\mathfrak{\hat g}$ corresponding to a Lie algebra $\mathfrak{g}$ is a central extension of the loop algebra $L\mathfrak{g}$. The $L\mathfrak{g}$ is the set of all Lie-Algebra valued functions on the circle, and is by a Fourier basis $t^a e^{i n \phi}$, where $t^a$ are the generators of $\mathfrak{g}$, and $\phi$ parameterizes the circle. This central extension gives rise to the short exact sequence
$$0 \rightarrow \mathbb{R} \rightarrow \mathfrak{\hat g} \rightarrow L\mathfrak{g} \rightarrow 0$$
Similarly, the Virasoro algebra $Vir$ is a central extension of the algebra of vector fields on a circle, $Vect(S^1)$, which is generated by the vector fields $i e^{i n\phi}\partial_\phi$. These give rise to a short exact sequence, Eq(4.13)
$$0 \rightarrow \mathbb{R} \rightarrow Vir \rightarrow Vect(S^1) \rightarrow 0$$
Let the central elements of the K-M algebra $\mathfrak{\hat g}$ be given by $K$, and that of $Vir$ by $C$.
The lecture notes state that $\mathfrak{\hat g}$ has an algebra
$$[t^a_m,t^b_n] = i f^{abc}t^c_{n+m} + \frac{1}{2}K n \delta^{ab}\delta_{n+m,0}$$
and that $Vir$ is given by
$$[l_m,l_n] = (n-m)l_{n+m} + \frac{1}{12} C (n^3-n) \delta_{n+m,0}$$
It's clear to me that all the terms not involving the central terms $K,C$ are correct, because those are just inherited from the original algebras. I'm confused as to how the other terms can be fixed on just an algebraic level. Any answers or references would be appreciated.
