Decomposition of involutions on inertially split division algebras (Q1586866)

Let \(F\) be a Henselian valued field with residue field \(\overline F\) of characteristic not~\(2\) and let \(S\) be an \(F\)-central (finite-dimensional) division algebra which is split by an inertial extension of \(F\). Assume \(\sigma\) is an involution on \(S\) (i.e., an anti-automorphism of period~\(2\)) which is the identity on an inertial lift in \(S\) of the center of the residue algebra \(\overline S\). The authors prove necessary and sufficient conditions for \(S\) to contain a \(\sigma\)-stable quaternion \(F\)-subalgebra or to decompose into a tensor product of \(\sigma\)-stable quaternion algebras, in terms of the decomposability of an associated central simple algebra over the residue field \(\overline F\). Their results extend to the general case of inertially split algebras over a Henselian field work of \textit{H.~Dherte} [Math. Z. 216, No. 4, 629-644 (1994; Zbl 0812.16023)] on Malcev-Neumann series division algebras.