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 $x_{1}, x_{2},…, x_{k}$ofsuchanalgebra$L$thereisfinite-dimensionalsubalgebra$A$whichcontainstheseelementsand$L(adA)^n\subset A$forsomeinteger$n$. 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.