Oriented pro-ℓ groups with the Bogomolov-Positselski property

For a prime number ℓ we say that an oriented pro-ℓ group (G, θ) has the Bogomolov–Positselski property if the kernel of the canonical projection on its maximal θ-abelian quotient πG,θab:G→G(θ) is a free pro-ℓ group contained in the Frattini subgroup of G. We show that oriented pro-ℓ groups of elementary type have the Bogomolov–Positselski property (cf. Theorem 1.2). This shows that Efrat’s Elementary Type Conjecture implies a positive answer to Positselski’s version of Bogomolov’s Conjecture on maximal pro-ℓ Galois groups of a field K in case that K×/(K×)ℓ is finite. Secondly, it is shown that for an H∙-quadratic oriented pro-ℓ group (G, θ) the Bogomolov–Positselski property can be expressed by the injectivity of the transgression map d22,1 in the Hochschild–Serre spectral sequence (cf. Theorem 1.4).