Involutions and Trace Forms on Exterior Powers of a Central Simple Algebra

For A a central simple algebra of degree 2n, the n-th exterior power algebra λ^nA is endowed with an involution which provides an interesting invariant of A. In the case where A is isomorphic to Q ⊗ B for some quaternion algebra Q, we describe this involution quite explicitly in terms of the norm form for Q and the corresponding involution for B