The level-one sl2 Sugawara central charge

The given brackets make ${\mathfrak{s}\mathfrak{l}}_2$ simple: $\mathrm{ad}(h)$ has the three distinct eigenspaces $\mathbb{C}e,\mathbb{C}f,\mathbb{C}h$; polynomial projections in $\mathrm{ad}(h)$ show any nonzero ideal contains one of $e,f,h$, and bracketing that vector with the other basis vectors yields all three. The $B$-dual basis to $(h,e,f)$ is $(h\mathrm{/}2,f,e)$, so ${\mathrm{\Omega }}_B=h^2\mathrm{/}2+ef+fe$. In the adjoint representation, $\mathrm{ad}({\mathrm{\Omega }}_B)e=\frac{1}{2}[h,[h,e]]+[e,[f,e]]+[f,[e,e]]=2e+2e+0=4e\mathrm{.}$ By [F2]–[F3], $2h^{\vee }=4$, hence $h^{\vee }=2$.

The dimension is three. [F1] and [F4] therefore give $c=1\cdot 3\mathrm{/}(1+2)=1$. The denominator is nonzero, so the example lies away from the critical level $k=-2$. This is a conditional module example: it computes the central charge for every level-one module meeting the theorem hypotheses with no unitarity or integrability requirement.