Trialitarian groups and the Hasse principle

Let F be a field of characteristic ≠ 2 such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} is of cohomological 2- and 3-dimension ≤ 2. For G a simply connected group of type 3D4 or 6D4 over F, we show that the natural map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}where ΩF is the set of orderings of F and Fv denotes the completion of F at v, restricts to be injective on the image of H1(F, Z(G)) in H1(F, G).