Structure Theory for One Class of Locally Finite Lie Algebras

In this paper I consider locally finite Lie algebras of characteristic zero satisfying the condition that for every finite number of elements $x_{1}, x_{2},..., x_{k}$ of such an algebra $L$ there is finite-dimensional subalgebra $A$ which contains these elements and $L(adA)^n\subset A$ for some integer $n$. For such algebras I prove several structure theorems that can be regarded as generalizations of the classical structure theorems of the finite-dimensional Lie algebras theory.