Generalized Poisson brackets and lie algebras of type H in characteristic 0

The Lie algebra of Cartan type H which occurs as a subalgebra of the Lie algebra of derivations of the polynomial algebra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $F[x_1,\dots,x_n, x_{-1},\dots,x_{-n}]$\end{document} was generalized by the first author to a class which included a subalgebra of the derivations of the Laurent polynomials \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $F[x_1,\dots,x_n,x_{-1}, \dots,x_{-n}, x_1^{-1},\dots,x_n^{-1},x_{-1}^{-1},\dots,x_{-n}^{-1}]$\end{document}. We show in this paper that these generalizations of Cartan type H algebras are isomorphic to certain generalizations of the classical algebra of Poisson brackets, and that it can be generalized further. In turn, these algebras can be recast in a form that is an adaption of a class of Lie algebras of characteristic p that was defined in 1958 be R. Block. A further generalization of these algebras is the main topic of this paper. We show when these algebras are simple, find their derivations, and determine all possible isomorphisms between two of these algebras.