Let $\mathfrak{g}$ be a semisimple Lie algebra. Define its loop algebra $L(\mathfrak{g})=\mathfrak{g}[t,t^{-1}]$ as the space of Laurent series of finite length with values in $\mathfrak{g}$. A typical element of $L(\mathfrak{g})$ is hence of the form $a(t) = \sum_{i}a_it^{i}$. Its derivative is denoted by $a'(t)$. The following is a standard theorem:
Theorem 1. The second cohomology of $L(\mathfrak{g})$ is 1-dimensional and is spanned by the 2-cocycle $$\alpha(a(t), b(t)) = Res_{t=0}(\langle a'(t), b(t)\rangle).$$
Question. Where can I find a reference that goes through the proof of this? The references I've been able to find so far only sketch out the proof or leave it as an exercise altogether.