In this paper they state the folowing: (for $\hat {\mathfrak g}$ a Kac Moody algebra $\mathfrak g\oplus c\mathbb C$)
I remember seeing this stated somewhere else before, but I cannot for the life of me find a text where they prove it (any of the two parts of the statement). So I would like to know where I can find a proof or at least an idea for one.