Let $G$ be a connected reductive group over $\mathbb{C}$. Let $F=\mathbb{C}((t))$ and $\bar{F}$ its algebraic closure. Let $c$ be a Cartan subalgebra of $g$ and $N$ its normalizer. It is written in *Fixed point varieties on affine flag manifolds* by Kazhdan and Lusztig that Cartan subalgebras of $G(F)$ are classified by $H^1(Gal(\bar{F}/F), N(\bar{F}))$. The map from the set of Cartan subalgebras to $H^1(Gal(\bar{F}/F), N(\bar{F}))$ is easy to construct, using the fact that Cartan subalgebras in $G(\bar{F})$ are conjugate to each other. My question is: How do I construct the map in the other direction? i.e. given $\sigma\in H^1(Gal(\bar{F}/F), N(\bar{F}))$, how do I construct a Cartan subalgebra of $G(F)$?



classifying non-split Cartan subalgebras
The field $\mathbb C((t))$ is $C\_1$ by a theorem of Lang (1952). This implies that $H^1(Gal(\bar F/F,G(\bar F))$ is trivial. Hence, for any 1-cocycle $n\_\gamma$ of $N(\bar F)$ there is $g\in G(\bar F)$ with $n\_\gamma=g^{-1}{}^\gamma g$. Then $\tilde c:=(\mathrm{Ad} g)(c)$ is a Cartan subalgebra which is defined over $F$:
$$
{}^\gamma \tilde c=(\mathrm{Ad} {}^\gamma g)({}^\gamma c)=(\mathrm{Ad}\ g n\_\gamma)(c)=\tilde c.
$$


