Decomposition of involutions on inertially split division algebras

Let F be a Henselian valued field with char((F)over bar) not equal 2, and let S be an inertially split F-central division algebra with involution sigma* that is trivial on an inertial lift in S of the field Z((S)over bar). We prove necessary and sufficient conditions for S to contain a sigma*-stable quaternion F-subalgebra, and for (S, sigma*) to decompose into a tensor product of quaternion algebras. These conditions are in terms of decomposability of an associated residue central simple algebra (I)over bar that arises from a Brauer roup decomposition of S.