  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 $\operatorname{cd}_2(F) \leq 2$, 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
$\operatorname{cd}_2(F) \leq 2$ as a tensor product of quaternion algebras.
