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 π: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. 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 in the Hochschild-Serre spectral sequence.