Artin–Springer property for type D groups under odd-degree extensions

Determine whether the Artin–Springer property extends to simple linear algebraic groups of type D: prove that for any field k and any odd-degree field extension L/k, every simple linear algebraic group of type D that is anisotropic over k remains anisotropic over L.

Background

The classical Artin–Springer theorem asserts that an anisotropic quadratic form over a field remains anisotropic under odd-degree field extensions. The natural analogue for groups of type D asks for preservation of anisotropy under odd-degree extensions. Up to isogeny, type D groups can be represented as groups of similitudes of algebras with quadratic pairs; in this setting, anisotropy corresponds to the anisotropy of an associated generalized quadratic form.

The general case of this property for type D groups is largely open. No counterexample is known, and various partial results exist, including positive results when the underlying central simple algebra has Schur index 2. This paper provides a new, characteristic-free proof for the index-2 case via generic splitting fields, but the full question for all type D groups remains unresolved.