Pfaffians, central simple algebras and similitudes (Q919439)

The aim of this paper is to give some applications of a pfaffian constructed in a previous paper [the authors, J. Reine Angew. Math. 398, 187-218 (1989; Zbl 0664.10008)]. The pfaffian of an Azumaya algebra A with involution of rank 16 is a quadratic form of rank 6, defined on the set of alternating elements, with value in a discriminant module pf(A). We show that two such algebras are isomorphic (as algebras) if and only if their pfaffians are similar. This generalizes a result of Jacobson on tensor products of quaternion algebras over fields of characteristic different from 2. Another application is the following generalization of a theorem of Albert. Let R be a commutative ring in which 2 is invertible. If A is an Azumaya algebra of rank 16 over R with an involution of orthogonal type and if the discriminant module pf(A) of A is trivial, then A is isomorphic, as an algebra with involution, to a tensor product of quaternion algebras. As a consequence, we get a necessary and sufficient condition for an involution of a rank 16 Azumaya algebra to admit an invariant quaternion subalgebra. We recall that there are examples of \textit{S. A. Amitsur}, \textit{L. H. Rowen} and \textit{J. P. Tignol} of involutions on central simple algebras of rank 16 which do not have any invariant quaternion algebras [Isr. J. Math. 33, 133-148 (1979; Zbl 0422.16010)]. In the last section we use the same techniques to compute the group of special similitudes of rank 6 quadratic spaces over commutative rings. This puts in a general set-up the classical computations of \textit{J. Dieudonné} [Acta Math. 87, 175-242 (1952; Zbl 0049.025)] for forms of dimension 6 over fields of characteristic not equal to 2. We also indicate a method of computation of the group of similitudes of rank 4 quadratic spaces.