Decomposition of involutions on inertially split division algebras

Abstract

Let F be a Henselian valued field with \(\mathrm{char}(\overline{F})\neq 2\), and let S be an inertially splitF}-central division algebra with involution $\sigma ^{\ast }$ that is trivial on an inertial lift in S of the field \(Z(\overline{S})\). We prove necessary and sufficient conditions for S to contain a \(\sigma ^{\ast }\)-stable quaternion {\it F}-subalgebra, and for \((S,\sigma ^{\ast })\) to decompose into a tensor product of quaternion algebras. These conditions are in terms of decomposability of an associated residue central simple algebra \(\overline{I}\) that arises from a Brauer group decomposition of S.