Identities related to a pair of generalized skew derivations on Lie ideals

Let $$\mathfrak {S}$$ S be a prime ring with $$char({\mathfrak {S}}) \ne 2$$ c h a r ( S ) ≠ 2 , $${\mathcal {Q}}_r$$ Q r its right Martindale quotient ring, $${\mathcal {C}}$$ C its extended centroid, L a non-central Lie ideal of $${\mathfrak {S}}$$ S , $${\mathcal {F}}$$ F and $${\mathcal {G}}$$ G two generalized skew derivations of $${\mathfrak {S}}$$ S . If $${\mathcal {F}}({\mathfrak {r}}{\mathfrak {s}})\pm {\mathcal {G}}({\mathfrak {s}}){\mathcal {G}}({\mathfrak {r}}) \pm {\mathfrak {s}}{\mathfrak {r}}=0$$ F ( r s ) ± G ( s ) G ( r ) ± s r = 0 , for any $${\mathfrak {r}},{\mathfrak {s}} \in L$$ r , s ∈ L , then one of the following holds: (1) $${\mathcal {F}}=0$$ F = 0 and there exists $$\lambda \in {\mathcal {C}}$$ λ ∈ C such that $${\mathcal {G}}({\mathfrak {r}})=\lambda {\mathfrak {r}}$$ G ( r ) = λ r , for any $${\mathfrak {r}}\in {\mathfrak {S}}$$ r ∈ S , with $$\lambda ^2+1=0$$ λ 2 + 1 = 0 ; (2) $${\mathfrak {S}}$$ S satisfies the standard polynomial identity $$s_4({\mathfrak {r}}_1,\ldots ,{\mathfrak {r}}_4)$$ s 4 ( r 1 , … , r 4 ) .