(1) We have explained above that Z G ( T d ) ≃ η H R. This last group is loop reductive by definition since η is a loop cocycle.
In the course of the proof of Theorem 12.1 we showed that up to conjugacy by an element of G ( k ), we can assume that σ normalizes the standard parabolic group P I and also the standard Levi subgroup L I. We have explained above that Z G ( T d ) ≃ η H R. This last group is loop reductive by definition since η is a loop cocycle.
Let η ∈ Z 1 ( π 1 ( X, a ), G ( k ¯ ) ) be a loop cocycle such that the twisted R n -group H = η ( G R n ) is anisotropic.
Let η : π 1 ( R n ) → H ( k ¯ ) be a loop cocycle and consider the loop reductive R n -groups H = η H R n and H ∘ = η H ∘ R n.
Let η ∈ Z 1 ( π 1 ( X, a ), G ( k ¯ ) ) be a loop cocycle and recall the decomposition η = ( η g e o, z ) into geometric and arithmetic parts described in Lemma 4.3.
Let H be the Chevalley k –form of H and let η : π 1 ( R n ) → Aut ( H ) ( k ¯ ) be a loop cocycle such that H = η ( H R n ).
π 1 ( X, a ) = Z ^ ( 1 ) n : In this case H 1 ( π 1 ( X, a ), G ( k ¯ ) ) is the set of conjugacy classes of n –tuples σ = ( σ 1, ⋯, σ n ) of commuting elements of finite order of G ( k ¯ ). 4. By means of the decompositions (4.1) and (4.2) we can think of loop cocycles as being comprised of a geometric and an arithmetic part, as we now explain.
They are given by (continuous) cocycles in the image of the natural map Z 1 ( π 1 ( X, a ), G ( k ¯ ) ) → Z e ´ t 1 ( X, G ), which we call loop cocycles.
The notion of loop torsor behaves well under twisting by a Galois cocycle z ∈ Z 1 ( Gal ( k ), G ( k ¯ ) ). Indeed the torsion map τ z - 1 : H e ´ t 1 ( X, G ) → H e t 1 ( X, z G ) maps loop classes to loop classes.
