On the Centralizers of Derivations with Engel Conditions

Let R be a noncommutative prime ring and d, δ two nonzero derivations of R. If δ([d(x), x]_n) = 0 for all x ∈ R, then char R = 2, d² = 0, and δ = αd, where α is in the extended centroid of R. As an application, if char R ≠ 2, then the centralizer of the set {[d(x), x]_n {pipe} x ∈ R} in R coincides with the center of R.