Derivations with annihilator conditions on Lie ideals in prime rings

Let R be a prime ring with characteristic different from two, d a derivation of R, L a noncentral Lie ideal of R, and a is an element of R. In the present paper, it is shown that if one of the following conditions holds: (i) a(d(uv)(s) - (uv)(t))(n) = 0, (ii) a(d(uv)(s) + (uv)(t))(n) = 0, (iii) a(d(uv)(s)-( vu)(t))(n) = 0 and (iv) a(d(uv)(s) +(vu)(t))(n) = 0 for all u, v is an element of L, where n, s, t are fixed positive integers, then a = 0 unless R satisfies s(4), the standard polynomial identity in four variables.