Note on Lie ideals with symmetric bi-derivations in semiprime rings

Let R be a semiprime ring, U a square-closed Lie ideal of R and $$D:R\times R\rightarrow R$$ D : R × R → R a symmetric bi-derivation and d be the trace of D. In the present paper, we prove that the R contains a nonzero central ideal if any one of the following holds: i) $$d\left( x\right) y\pm xg(y)\in Z,$$ d x y ± x g ( y ) ∈ Z , ii) $$[d(x),y]=\pm [x,g(y)],$$ [ d ( x ) , y ] = ± [ x , g ( y ) ] , iii) $$d(x)\circ y=\pm x\circ g(y),$$ d ( x ) ∘ y = ± x ∘ g ( y ) , iv) $$[d(x),y]=\pm x\circ g(y),$$ [ d ( x ) , y ] = ± x ∘ g ( y ) , v) $$d([x,y])=[d(x),y]+[d(y),x],$$ d ( [ x , y ] ) = [ d ( x ) , y ] + [ d ( y ) , x ] , vi) $$d(xy)\pm xy\in Z,$$ d ( x y ) ± x y ∈ Z , vii) $$d(xy)\pm yx\in Z,$$ d ( x y ) ± y x ∈ Z , viii) $$d(xy)\pm [x,y]\in Z,$$ d ( x y ) ± [ x , y ] ∈ Z , ix) $$d(xy)\pm x\circ y\in Z,$$ d ( x y ) ± x ∘ y ∈ Z , x) $$g(xy)+d(x)d(y)\pm xy\in Z,$$ g ( x y ) + d ( x ) d ( y ) ± x y ∈ Z , xi) $$g(xy)+d(x)d(y)\pm yx\in Z,$$ g ( x y ) + d ( x ) d ( y ) ± y x ∈ Z , xii) $$g([x,y])+[d(x),d(y)]\pm [x,y]\in Z,$$ g ( [ x , y ] ) + [ d ( x ) , d ( y ) ] ± [ x , y ] ∈ Z , xiii) $$g(x\circ y)+d(x)\circ d(y)\pm x\circ y\in Z,$$ g ( x ∘ y ) + d ( x ) ∘ d ( y ) ± x ∘ y ∈ Z , for all $$x,y\in U,$$ x , y ∈ U , where $$G:R\times R\rightarrow R$$ G : R × R → R is symmetric bi-derivation such that g is the trace of G.