Banach Lie algebras with Lie subalgebras of finite codimension have Lie ideals

This paper continues the work started in [E. Kissin, V. S. Shulman and Yu. V. Turovskii, 'Banach Lie algebras with Lie subalgebras of finite codimension: their invariant subspaces and Lie ideals', J. Funct. Anal. 256 (2009) 323-351.] and is devoted to the study of reducibility of an infinite-dimensional Lie algebra of operators on a Banach space when its Lie subalgebra of finite codimension has an invariant subspace of finite codimension. In addition to the tools developed in the above paper; filtrations of Banach spaces with respect to Lie algebras of operators and related systems of operators on graded Banach spaces, the present paper introduces and studies some new concepts and techniques: the theory of Lie quasi-ideals and properties of Lie nilpotent finite-dimensional subspaces of Banach associative algebras. The application of these techniques to an operator Lie algebra L shows that, under some mild additional assumptions, L is reducible if its Lie subalgebra of finite codimension has an invariant subspace of finite codimension. This, in turn, leads to the main result of the paper: if a Banach Lie algebra L has a closed Lie subalgebra of finite codimension, then it has a proper closed Lie ideal of finite codimension. Moreover, if L is non-commutative, then it has a characteristic Lie ideal of finite codimension, that is, a proper closed Lie ideal of L invariant for all bounded derivations of L.