Generalized Cartan type K Lie algebras in characteristic 0

The Lie algebra of Cartan type K which occurs as a subalgebra of the Lie algebra of derivations of the polynomial algebra F[x(0),x(1),...,x(n),x(-1),...,x(-n)], where F is a field of characteristic 0, was generalized by the first author to a class which included a subalgebra of the derivations of the Laurent polynomials F[x(0),x(1),...,x(n),x(-1),...,x(-n),x(0)(-1),x(1)(-1),...,x(n)(-1),x(-1)(-1),...,x(-n)(-1)]. A further generalization of these algebras is the main topic of this paper. We show when these algebras are simple, determine all possible isomorphisms between two of these algebras and all possible isomorphisms between a generalized Cartan type K Lie algebra and a generalized Cartan type H Lie algebra.