Decomposition of involutions on inertially split division algebras

Abstract

Let F be a Henselian valued field with char(F ) = 2, and let S be an inertially split F -central division algebra with involution σ * that is trivial on an inertial lift in S of the field Z(S). We prove necessary and sufficient conditions for S to contain a σ * -stable quaternion F -subalgebra, and for (S, σ * ) to decompose into a tensor product of quaternion algebras. These conditions are in terms of decomposability of an associated residue central simple algebra I that arises from a Brauer group decomposition of S.