Involutions on tensor products of quaternion algebras

We study possible decompositions of totally decomposable algebras with involution, that is, tensor products of quaternion algebras with involution. In particular, we are interested in decompositions in which one or several factors are the split quaternion algebra $M_2(F)$, endowed with an orthogonal involution. Using the theory of gauges, developed by Tignol-Wadsworth, we construct examples of algebras isomorphic to a tensor product of quaternion algebras with $k$ split factors, endowed with an involution which is totally decomposable, but does not admit any decomposition with $k$ factors $M_2(F)$ with involution. This extends an earlier result of Sivatski where the algebra considered is of degree $8$ and index $4$, and endowed with some orthogonal involution.