  We show that finitely subgraded Lie algebras of compact operators have
invariant subspaces when conditions of quasinilpotence are imposed on certain
components of the subgrading. This allows us to obtain some useful information
about the structure of such algebras. As an application, we prove a number of
results on the existence of invariant subspaces for algebraic structures of
compact operators. Along the way we obtain new criteria for the
triangularizability of a Lie algebra of compact operators.
