THE GENERALIZED EVEN CLIFFORD-ALGEBRA

For a central simple algebra A with an orthogonal involution * over a field F of characteristic not equal to two, we associate to it the generalized even Clifford algebra C-0(A, *). We will present results on the structure of this algebra C-0(A, *), some of which are due to Jacobson in (J. Algebra 1 (1964), 288-300). Using a generic method, we then give a formula for computing C-0(A, *) when (A, *) is decomposable as algebras with involution. Using these results, a necessary condition for decomposability of an algebra with orthogonal involution compatible with the involution is then obtained. (C) 1995 Academic Press, Inc.