Nonabelian $H^1(\Gamma,G)$ and 1-Cocycles

Let $\Gamma=\mathrm{Gal}(K/k)$ act on $G(K)$ by field automorphisms.

A 1-cocycle is a map $a:\Gamma\to G(K)$ satisfying $a_{\sigma\tau}=a_\sigma\cdot {}^\sigma a_\tau.$

Two cocycles are cohomologous if $a'_\sigma=g^{-1}a_\sigma\,{}^\sigma g$ for some $g\in G(K)$; the set of classes is nonabelian cohomology $H^1(\Gamma,G)$ (a pointed set).

In the letter: an “inner twisting” is specified by such a cocycle $\{a_\tau\}$, and “splitting locally at almost all $p$” means the restriction class becomes trivial for local decomposition groups.