In this paper, we determine the isomorphism classes of contact simple Lie algebras constructed from locally-finite commuting derivations, introduced earlier by the second author. These algebras are in general not finitely graded.

The present paper is another step toward the classification of Z × Z–graded Lie algebras: we classify Z × Z–graded Lie algebras A =⊕i,j∈Z Ai,j over a field F of characteristic 0 with dimAi,j ≤ 1 for i, j ∈ Z, satisfying: (1) A0 = ⊕t∈Z Ai,0 Vir, the centerless Virasoro algebra, (2) A0,−1, A0,0, A0,1 span the 3-dimensional simple Lie algebra sl2, (3) A is generated by the centerless Virasoro algebra and sl2. These algebras include some rank 2 centerless Virasoro algebras, and some rank 2 Block algebras and their extensions.

n one of our recent papers, the associative and the Lie algebras of Weyl type A[D] = AUF[D] were de ned and studied, where A is a commutative associative algebra with an identity element over a eld F of any characteristic, and F[D] is the polynomial algebra of a commutative derivation subalgebra D of A. In the present paper, a class of the above associative and Lie algebras A[D] with F being a eld of characteristic 0, D consisting of locally nite but not locally nilpotent derivations of A, are studied. The isomorphism classes and automorphism groups of these associative and Lie algebras are determined.

In a recent paper by Xu, some simple Lie algebras of generalized Cartan type were constructed, using the mixtures of grading operators and down-grading operators. Among them are the simple Lie algebras of generalized Witt type, which are in general nongraded and have no torus. In this paper, some representations of these simple Lie algebras of generalized Witt type are presented.