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The paper develops the theory of topological radicals of Banach Lie algebrasand studies the structure of Banach Lie algebras with sufficiently many Liesubalgebras of finite codimensions -- the intersection of all these subalgebrasis zero. It is shown that the intersections of certain families of Liesubalgebras (closed Lie subalgebras of finite codimension, closed Lie ideals offinite codimension, closed maximal Lie subalgebras of finite codimension,closed maximal Lie ideals of finite codimension) correspond to differentpreradicals, and that these preradicals generate the same radical, the Frattiniradical. The main attention is given to structural properties ofFrattini-semisimple Banach Lie algebras and, in particular, to a newinfinite-dimensional phenomenon associated with the strong Frattini preradicalintroduced in this paper. A constructive description of Frattini-free BanachLie algebras is obtained.

Topological radicals, IV. Frattini theory for Banach Lie algebras