In  we proved the following isotropy-reflection principle:
Theorem. Let F be a formally real field and let Fp denote its Pythagorean closure. The natural embedding of reduced special groups from G
red(F) into G
(Fp) = G(FP) induced by the inclusion of fields, reflects isotropy.
red(F) denotes the reduced special group (with underlying group Ḟ/ΣḞ2) associated to the field F, henceforth assumed formally real; cf. , Chapter 1, §3, for details.
The result proved in  is, in fact, more general. For example, the Pythagorean closure Fp
can be replaced in the statement above by the intersection of all real closures of F (inside a fixed algebraic closure). Similar statements hold, more generally, for all relative Pythagorean closures of F in the sense of Becker , Chapter II, §3.
Since the notion of isotropy of a quadratic form can be expressed by a first-order formula in the natural language LSG
for special groups (with the coefficients as parameters), this result raises the question whether the embedding ιFFp: G
red(F) ↪ G (Fp
) is elementary. Further, since the LSG
-formula expressing isotropy is positive-existential, one may also ask whether ιFFp
reflects all (closed) formulas ofthat kind with parameters in G
In this paper we give a negative answer to the first of these questions, for a vast class of formally real (non-Pythagorean) fields F (Prop. 5.1). This follows from rather general preservation results concerning the “Boolean hull” and the “reduced quotient” operations on special groups.