A class of infinite dimensional simple Lie algebras

Let A be an abelian group, F be a field of characteristic 0, and alpha, beta be linearly independent additive maps from A to F, and let delta epsilon ker(alpha)\{0}. Then there is a Lie algebra L = L(A, alpha, beta, delta) = circle plus(x epsilon A) Fe-x under the product [e(x), e(y)] = alpha(x - y) e(x+y)+(alpha boolean AND beta) (x, y) e(x+y-delta). If, further, beta(delta) = 1, and beta(A) = Z, there is a subalgebra L+ - L(A(+), alpha,beta,delta) = circle plus(x epsilon A) + Fe-x, where A(+) = {x epsilon A / beta(x) greater than or equal to 0} The necessary and sufficient conditions are given for L' = [L, L] and L+ to be simple, and all semi-simple elements in L' and Lf are determined. It is shown that L' and L+ cannot be isomorphic to any other known Lie algebras and L' is not isomorphic to any L+, and all isomorphisms between two L' and all isomorphisms between two L+ are explicitly described.