  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.
