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Factor of $1/2$ in the Sugawara construction

I'm trying to reproduce the Sugawara construction calculation using this reference (page 14). The normal-ordering of two local operators is defined as $$ N(XY)(w)=\frac{1}{2\pi i} \oint_w \frac{dx}{...
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OPE Kac-Moody Currents

We have the following operators: \begin{align} J^a(z) = \frac{1}{2}\psi_s^{\dagger}(z)\sigma^a_{s s'}\psi_{s'}(z), \hspace{10 mm} \bar{J}^a(z) = \frac{1}{2}\psi_s^{\dagger}(\bar{z})\sigma^a_{s s'}\...
2
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Kac-Moody algebra, proof of parameters calculation

I'm following the notes "Ginsparg - Applied Conformal Field Theory" (https://arxiv.org/abs/hep-th/9108028) and I'm stuck on a proof at page 140 about Kac-Moody algebras. I would like to prove that $\...