Generalized ?-Lie ideal of ?-prime ring

Let R be a ∗-prime ring with characteristic not 2, σ, τ : R → R be two automorphisms, U be a nonzero ∗- (σ, τ ) -Lie ideal of R such that τ commutes with ∗, and a, b be in R. (i) If a ∈ S∗ (R) and [U, a] = 0, then a ∈ Z (R) or U ⊂ Z (R). (ii) If a ∈ S∗ (R) and [U, a]σ,τ ⊂ Cσ,τ , then a ∈ Z (R) or U ⊂ Z (R). (iii) If U ̸⊂ Z (R) and U ̸⊂ Cσ,τ , then there exists a nonzero ∗-ideal M of R such that [R, M]σ,τ ⊂ U but [R, M]σ,τ ̸⊂ Cσ,τ . (iv) Let U ̸⊂ Z (R) and U ̸⊂ Cσ,τ . If aUb = a ∗Ub = 0, then a = 0 or b = 0.