Sugawara construction ($L_m=(2k+Q{)}^{-1}{\sum }_{a,p}:J_{m-p}^a{J}_p^a:$)

For a restricted representation of an affine current algebra with $[J_m^a,J_n^b]=i{f}^{abc}{J}_{m+n}^c+km{\delta }^{ab}{\delta }_{m+n,0}$ and $f^{abc}{f}^{abd}=Q{\delta }^{cd}$, the displayed normal ordering construction is defined when $2k+Q\ne 0$. Moving the currents past each other gives $[L_m,J_n^a]=-n{J}_{m+n}^a$; the additional $Q$ in the denominator is the adjoint-Casimir reordering term. The resulting Virasoro algebra has central charge $c=k\dim \mathfrak{g}/(k+Q/2)$. On a compact unitary positive-energy module, $L_m^{†}=L_{-m}$.

For a current subalgebra $\mathfrak{h}\subset \mathfrak{g}$, its currents have conformal weight one under both Sugawara construction stress tensors. Their difference therefore commutes with all $\mathfrak{h}$ currents and with the $\mathfrak{h}$ stress tensor. It obeys the Virasoro algebra with central charge $c_{\mathfrak{g}}-c_{\mathfrak{h}}$. For a positive-level compact unitary vacuum representation, $\Vert L_{-2}^{\mathfrak{g}/\mathfrak{h}}|0⟩{\Vert }^2=(c_{\mathfrak{g}}-c_{\mathfrak{h}})/2\ge 0$. A Cartan subalgebra has $c_{\mathfrak{h}}=\mathrm{rank}\mathfrak{g}$, giving the lower bound on $c_{\mathfrak{g}}$ under these hypotheses.