The twisted Weyl-vector formula for twisted loop algebras

Let $\widetilde{\mathfrak{g}}$ be a twisted loop algebra with $\epsilon(\mathfrak{h})=0$. Let $\widehat{\rho}$ be the Weyl vector determined by the simple roots of $\widetilde{\mathfrak{g}}$ through

$$2(\widehat{\rho}\mid\widehat{\alpha_i})=(\widehat{\alpha_i}\mid\widehat{\alpha_i}).$$

For each positive root $\alpha\in\Delta_+$, write $p_\alpha$ for its parity and let $\epsilon_\alpha\equiv\epsilon(u_\alpha)$. Twisted Weyl-vector formula. The Weyl vector is

$$\widehat{\rho}=(\widetilde{\rho},h^\lor,0),$$

where

$$\widetilde{\rho}=\frac{1}{2}\sum_{\alpha\in\Delta_+}(-1)^{p_\alpha}\alpha(1-2\epsilon_\alpha).$$

This identifies the affine Weyl vector in the twisted case with a finite-dimensional “twisted rho” together with its $h^\lor$ and $D$ components; the source provides no resolution or further evidence, so the claim is left open.