Galois Descent, Twisted Forms, and Inner Forms

Let $K/k$ be a finite Galois extension with group $\Gamma=\mathrm{Gal}(K/k)$ and let $G$ be a split $k$-group.

A homomorphism $\delta:\Gamma\to \mathrm{Aut}(G)$ (often landing in a pinned automorphism group like $\Omega$) defines a twisted form $G_\delta$ over $k$ by descent from $G_K$ with $\Gamma$-action twisted by $\delta$.

An inner twist is specified by a (non-abelian) 1-cocycle $a:\Gamma\to G(K)$ satisfying $a_{\sigma\tau}=a_\sigma\cdot {}^\sigma a_\tau$; cocycles up to coboundary form $H^1(\Gamma,G)$.

In the letter: $G$ is built as a $\delta$-twist followed by an inner twist; at almost all primes this inner cocycle “splits” locally.