Power values of generalized skew derivations preserving Jordan product on Lie ideals

Let R be a non-commutative prime ring of characteristic different from 2, Qr its right Martindale quotient ring and C its extended centroid. Suppose that L is a non-central Lie ideal of R, F a non-zero generalized skew derivation of R and (Formula presented.) a fixed integer. If (Formula presented.) for all (Formula presented.) then one of the following holds: R satisfies (Formula presented.) the standard polynomial identity on four non-commuting variables; there exists (Formula presented.) such that (Formula presented.) for all (Formula presented.) with (Formula presented.).