Abstract
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 algebraC0(A, ∗). We will present results on the structure of this algebra C0(A, ∗), some of which are due to Jacobson in (J. Algebra1 (1964), 288-300). Using a generic method, we then give a formula for computing C0(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.
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