Square-central and Artin–Schreier elements in division algebras

We study the behavior of square-central elements and Artin–Schreier elements in division algebras of exponent 2 and degree a power of 2. We provide chain lemmas for such elements in division algebras over 2-fields F of cohomological 2-dimension cd2(F)≤2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm cd}_2(F) \leq 2}$$\end{document} and deduce a common slot lemma for tensor products of quaternion algebras over such fields. We also extend to characteristic 2 a theorem proven by Merkurjev for characteristic not 2 on the decomposition of any central simple algebra of exponent 2 and degree a power of 2 over a field F with cd2(F)≤2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm cd}_2(F) \leq 2}$$\end{document} as a tensor product of quaternion algebras.