Sugawara embeddings give a Virasoro coset stress tensor

For $x,y\in \mathfrak{h}$, the induced current bracket is the sum of the brackets in each simple factor. Since each ${\iota }_i$ preserves brackets and distinct factors commute, $[J_x^{\mathfrak{h}}(s),J_y^{\mathfrak{h}}(t)]=J_{[x,y]}^{\mathfrak{h}}(s+t)+s({\sum }_i{I}_i{k}_i)B_{\mathfrak{h}}(x,y){\delta }_{s+t,0}I$. Thus the induced level is $\mathrm{\ell }$.

Apply [F6] to every ${\mathfrak{g}}_i$ action and to the restricted $\mathfrak{h}$ action, using the noncritical assumptions and the normalization [F4]. This gives each factor and denominator their Sugawara Virasoro brackets and central charges $c_i=k_i\dim {\mathfrak{g}}_i\mathrm{/}(k_i+h_i^{\vee })$ and $c_{\mathfrak{h}}=\mathrm{\ell }\dim \mathfrak{h}\mathrm{/}(\mathrm{\ell }+h_{\mathfrak{h}}^{\vee })$.

By [A1], modes from distinct numerator factors commute. Adding their Virasoro brackets from step 1.2 shows that $L^{\mathfrak{g}}={\sum }_i{L}^{(i)}$ is Virasoro with $c_{\mathfrak{g}}={\sum }_i{c}_i$.

Applying [F5] to each numerator action and to the induced restricted denominator action gives $[L_m^{\mathfrak{g}},J_x^{\mathfrak{h}}(n)]=-n{J}_x^{\mathfrak{h}}(m+n)$ and $[L_m^{\mathfrak{h}},J_x^{\mathfrak{h}}(n)]=-n{J}_x^{\mathfrak{h}}(m+n)$. The other-factor commutators vanish by [A1], so summing the component identities gives the numerator formula; the denominator identity uses its level $\mathrm{\ell }$ from step 1.1.

Subtract the two current-action identities from step 2.2. For every $x\in \mathfrak{h}$ and $m,n\in \mathbb{Z}$, this gives $[L_m^{\mathfrak{g}\mathrm{/}\mathfrak{h}},J_x^{\mathfrak{h}}(n)]=0$.

Put $q_{\mathfrak{h}}=\mathrm{\ell }+h_{\mathfrak{h}}^{\vee }$, let $(e_a)$ be a $B_{\mathfrak{h}}$-orthonormal basis, and write $N_r(A,B)=AB$ for $r<0$ and $N_r(A,B)=BA$ for $r\ge 0$. The product rule and the current action in step 2.2 give $2q_{\mathfrak{h}}[L_m^{\mathfrak{g}},L_n^{\mathfrak{h}}]={\sum }_{a,r}(-r{N}_r(J_{e_a}^{\mathfrak{h}}(m+r),J_{e_a}^{\mathfrak{h}}(n-r))-(n-r)N_r(J_{e_a}^{\mathfrak{h}}(r),J_{e_a}^{\mathfrak{h}}(m+n-r)))\mathrm{.}$ This identity is valid on each vector: the original sums stabilize by [F1], and restrictedness [F2] kills both tails after the commutator expansion.

Set $p=m+n$, reindex the first sum by $s=m+r$, and combine terms with the same normal order $N_s$. Their coefficient is $(m-s)-(n-s)=m-n$, giving $2q_{\mathfrak{h}}(m-n)L_p^{\mathfrak{h}}$. The only order correction is ${\sum }_s(m-s)f_m(s){\sum }_a[J_{e_a}^{\mathfrak{h}}(s),J_{e_a}^{\mathfrak{h}}(p-s)]$, where $f_m(s)={\mathbf{1}}_{s<m}-{\mathbf{1}}_{s<0}$. This sum is finite because $f_m$ is supported between $0$ and $m$.

By step 1.1 and orthonormality, ${\sum }_a[J_{e_a}^{\mathfrak{h}}(s),J_{e_a}^{\mathfrak{h}}(p-s)]=s\mathrm{\ell }\dim (\mathfrak{h}){\delta }_{p,0}I$, since $[e_a,e_a]=0$. For $m>0$, ${\sum }_{s}s(m-s)f_m(s)={\sum }_{s=0}^{m-1}s(m-s)=(m^3-m)\mathrm{/}6$. For $m=-u<0$, substituting $s=-t$ gives $-{\sum }_{t=1}^{u}t(u-t)=-(u^3-u)\mathrm{/}6=(m^3-m)\mathrm{/}6$; for $m=0$ the sum is zero.

Substituting step 5.1 into step 4.1 and dividing by $2q_{\mathfrak{h}}\mathrm{\ne }0$ yields $[L_m^{\mathfrak{g}},L_n^{\mathfrak{h}}]=(m-n)L_{m+n}^{\mathfrak{h}}+\frac{c_{\mathfrak{h}}}{12}(m^3-m){\delta }_{m+n,0}I$, using the value of $c_{\mathfrak{h}}$ from step 1.2. Antisymmetry gives the same mixed bracket for $[L_m^{\mathfrak{h}},L_n^{\mathfrak{g}}]$.

Step 6.1 is exactly the Virasoro bracket of $L^{\mathfrak{h}}$ from step 1.2, so subtraction gives $[L_m^{\mathfrak{g}\mathrm{/}\mathfrak{h}},L_n^{\mathfrak{h}}]=0$ for all $m,n$.

Expand $[L_m^{\mathfrak{g}}-L_m^{\mathfrak{h}},L_n^{\mathfrak{g}}-L_n^{\mathfrak{h}}]$. The two mixed brackets from step 6.1 subtract twice the $\mathfrak{h}$ bracket, while the separate $\mathfrak{h}$ Virasoro bracket from step 1.2 adds it once. The result is $(m-n)(L_{m+n}^{\mathfrak{g}}-L_{m+n}^{\mathfrak{h}})+\frac{c_{\mathfrak{g}}-c_{\mathfrak{h}}}{12}(m^3-m){\delta }_{m+n,0}I$, which is the claimed Virasoro relation and central charge.