Decomposition of involutions on inertially split division algebras

Let F be a Henselian valued field with , 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 . We prove necessary and sufficient conditions for S to contain a -stable quaternion {it F}-subalgebra, and for to decompose into a tensor product of quaternion algebras. These conditions are in terms of decomposability of an associated residue central simple algebra that arises from a Brauer group decomposition of S. Received February 1, 1999; in final form August 26, 1999 / Published online July 3, 2000