Sugawara modes satisfy Virasoro at every noncritical complex level

Choose a $B$-orthonormal basis $(e_a)$ and abbreviate $a_r=J_{e_a}(r)$. Existence follows by finite diagonalization of the nondegenerate symmetric complex form, rescaling each nonzero diagonal entry by a complex square root. Put $p=m+n$ and let $N_r(A,B)$ mean $AB$ for $r<0$ and $BA$ for $r\ge 0$; this subscript specifies the split, not an intrinsic index of $A$. Expanding by [F2] gives $2q[L_m,L_n]={\sum }_{a,r}(-r{N}_r(a_{m+r},a_{n-r})-(n-r)N_r(a_r,a_{p-r}))\mathrm{.}$ This equality is algebraic on each $v\in V$. Truncate the original quadratic sum at $\mathrm{\mid }r\mathrm{\mid }\le R$; it stabilizes on $v$ and $L_{m}v$ by [F1], so its commutator stabilizes to the left side. On the right, in the positive tail the rightmost indices are $r$ or $m+r$, and in the negative tail they are $n-r$ or $p-r$; [F4] kills all these tails for large $R$. Thus the displayed sum is locally finite on $v$, and fixed reindexings below do not create limiting errors.

Reindex the first term by $s=m+r$. It becomes $(m-s)N_{s-m}(a_s,a_{p-s})$, whereas the desired normal order is $N_s$. The second term already has the latter order. Their normally ordered coefficients combine to $(m-s)-(n-s)=m-n$. The order difference in the first term is $N_{s-m}(a_s,a_{p-s})-N_s(a_s,a_{p-s})=f_m(s)[a_s,a_{p-s}],f_m(s)={\mathbf{1}}_{s<m}-{\mathbf{1}}_{s<0}\mathrm{.}$ For $m>0$ it is the commutator for $0\le s<m$; for $m<0$ it is its negative for $m\le s<0$; for $m=0$ it is zero. Therefore $2q[L_m,L_n]=2q(m-n)L_p+{\sum }_s(m-s)f_m(s){\sum }_a[a_s,a_{p-s}]\mathrm{.}$ Every correction sum is finite, including all zero-index endpoints.

By [F3], $[a_s,a_{p-s}]=sk{\delta }_{p,0}{I}_V$, since $[e_a,e_a]=0$ and $B(e_a,e_a)=1$. The correction in step 2.1 is consequently $k\dim (\mathfrak{g}){\delta }_{p,0}{I}_V$ times ${\sum }_{s}s(m-s)f_m(s)$. For $m>0$, the finite sum is $m{\sum }_{s=0}^{m-1}s-{\sum }_{s=0}^{m-1}{s}^2=(m^3-m)\mathrm{/}6$. For $m=-u<0$, put $s=-t$ in the negative interval; it becomes $-{\sum }_{t=1}^{u}t(u-t)=-(u^3-u)\mathrm{/}6=(m^3-m)\mathrm{/}6$. At $m=0$ both sides vanish. These finite arithmetic identities follow by induction from ${\sum }_{s=0}^{d-1}s=d(d-1)\mathrm{/}2$ and ${\sum }_{s=0}^{d-1}{s}^2=d(d-1)(2d-1)\mathrm{/}6$. Dividing step 2.1 by $2q\mathrm{\ne }0$ gives exactly the stated central charge. The resulting brackets and the scalar central action satisfy [F5], hence give the claimed Virasoro representation, including when $V=0$.