Abstract
We study totally decomposable symplectic and unitary involutions on central simple algebras of index 2 and on split central simple algebras respectively. We show that for every field extension, these involutions are either anisotropic or hyperbolic after extending scalars, and that the converse holds if the algebras are of 2-power degree. These results are new in characteristic 2, otherwise the symplectic result was shown in Becher (Invent Math 173(1):1–6, 2008) and the unitary result was partly shown in Black and Queguiner-Mathieu (Enseign Math 60(3–4):377–395, 2014).
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