If $L$ is a finite-dimensional simple Lie algebra (over $\mathbb{C}$), then it is apparently the case that $H^2(L\otimes\mathbb{C}[t,t^{-1}],\mathbb{C})$ ($L\otimes\mathbb{C}[t,t^{-1}]$ being the loop algebra of $L$, with Lie bracket $[x\otimes f,y\otimes g]=[x,y]\otimes fg$) is isomorphic to $\mathbb{C}$. In the literature that I have looked at I can only find one proof (Conformal Field Theory and Topology - T. Kohno), but this relies on the fact that $L$ is the Lie algebra of a compact Lie group.
My question: is there an algebraic proof, and if so, does someone know or have a good reference that I can use?