Coset construction

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.