On embedded products of Grassmannians

Let Γ′ and Γ be two Grassmannians. The standard embedding φ : Γ′ × Γ→ P is obtained by combining the Plücker and Segre embeddings. Given a further embedding η : Γ′ × Γ→ P′, we find a sufficient condition for the existence of α ∈ Aut(Γ) and of a collineation ψ : P→ P′ such that η = (idΓ′ × α)φψ. A.M.S. classification number: 51M35.