I'm reading Rapoport's paper "On the classification and specialization of Fisocrystals with additional structure", and I'm confused for one detail. Here G is a reductive group over $F$, $F$ is a finite extension of $\mathbb{Q}_p$, L is the completion of the maximal unramified extension $F^{un}$, $\sigma$ is the Frobenius automorphism of L/F, $\Gamma =Gal(\bar{F}/F)$.
He sets $$ \mathcal{N}(G):= \text{(Int G(L)}\text{\Hom}_{L} (\mathbb{D}, G) )^{\sigma} $$where $\mathbb{D}$ is a pro-algebraic group over F whose character group is $\mathbb{Q}$.
Then he says that if $T\subset G$ is a maximal torus, then $$\mathcal{N}(G)= (X_{*}(T)_{\mathbb{Q}}/W )^{\Gamma} $$
I'm confused about it, I have read similar results in Kottwitz's paper "Shimura varieties and twisted orbital integral"
but this lemma assumes that $G$ is quasi-split. I have already known that it is right when $G$ is quasi-split, but I don't know if it is when $G$ is not quasi-split.