The shifted integrable character numerator is Weyl skew

Fix $i$ and write $E=e_i,F=f_i,H=h_i$. By F2 and F5 the module has the local nilpotence in F3, while F8 and F9 give $V\in \mathcal{O}$ and hence well-defined finite-dimensional weight spaces. Thus $T=\exp (E)\exp (-F)\exp (E)$ is defined on each vector by finite sums. Its inverse is $\exp (-E)\exp (F)\exp (-E)$: each adjacent exponential cancellation is the finite binomial identity on a vector. F4 permits all computations involving $E,F,H$ on a finite-dimensional invariant subspace containing the vector under consideration.

The brackets in F7 give conjugations $\exp (E)H\exp (-E)=H-2E$, $\exp (-F)H\exp (F)=H-2F$ and $\exp (-F)E\exp (F)=E+H-F$. These follow by expanding the finite exponentials, or by differentiating their polynomial conjugations and using the first two commutators. Consequently the successive conjugations of $H$ in $TH{T}^{-1}$ are $H-2E$, then $-H-2E$, then $-H$. For an arbitrary $h\in \mathfrak{h}$, put $h^{\perp }=h-{\alpha }_i(h)H\mathrm{/}2$. F7 makes $h^{\perp }$ commute with $E,F$, so $Th{T}^{-1}=h-{\alpha }_i(h)H=s_{i}h$. The polynomial identities hold on every vector by step 1.1; commutation with $h^{\perp }$ requires no finite-dimensional invariant space for the whole Cartan.

If $v\in V_{\mu }$, step 2.1 and $s_i^2=1$ give $hTv=T(s_{i}h)v=\mu (s_{i}h)Tv$. Thus $T:V_{\mu }\to V_{s_i\mu }$ is an isomorphism, with the inverse from step 1.1. The two finite dimensions in F8 are equal, including when both spaces are zero. Hence $s_i\mathrm{ch}V=\mathrm{ch}V$ as coefficient arrays.

Transform the coefficient sum for $D\mathrm{ch}V$ by $s_i$. Relabelling its pairs of exponents bijectively gives the coefficient sum for $(s_{i}D)(s_i\mathrm{ch}V)$; it is finite because its preimage is a coefficient sum of the original product, finite by F8. F1 and step 3.1 therefore give $s_{i}N=-N$. Iterating a finite word gives the claimed determinant sign. No assertion of an action on arbitrary completion elements is needed. All exponential identities were finite on each vector and no bases of infinitely many spaces were selected.