For a prime number ℓ we say that an oriented pro-ℓ group (G, θ) has the Bogomolov property if the kernel of the canonical projection on its maximal θ-abelian quotient π ab G,θ : G → G(θ) is a free pro-ℓ group contained in the Frattini subgroup of G. We show that pro-ℓ groups of elementary type have the Bogomolov 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 property can be expressed by the injectivity of the transgression map d 2,1 2 in the Hochschild-Serre spectral sequence (cf. Theorem 1.4).Then the maximal pro-ℓ Galois group of ℓ ∞ √ K is a free pro-ℓ group.A profinite group G admits a maximal pro-ℓ quotient G(ℓ) = G/O ℓ (G), where O ℓ (G) is the closed normal subgroup of G being generated by all pro-q Sylow