Z-graded identities of infinite dimensional Lie algebras in characteristic 2: revisited

Let K be any field of characteristic two and let U₁ U 1 and W₁ W 1 be the Lie algebras of the derivations of the algebra of Laurent polynomials K[t,t⁻¹] K [ t , t - 1 ] and of the polynomial ring K [ t ], respectively. The algebras U₁ U 1 and W₁ W 1 are equipped with their natural Z Z -gradings. The graded identities of these two algebras were studied and described in [1]. The main goal of this paper is to present streamlined versions of Theorems 2.1 and 2.2 from [1] when the field K is finite.