What is the outer automorphism group of the complex Lie algebra of bounded operators on a complex Hilbert space, with the commutator as Lie bracket? What for the real Lie algebra of bounded antihermitian operators? Does their structure depend on whether the space is finite-dimensional, infinite-dimensional separable, or inseparable? Does it depend on continuity assumptions in appropriate topologies?
There are obvious subgroups, namely the group of inner automorphisms given by conjugation by some invertible element of the algebra in the first case, and by conjugation by a unitary or antiunitary operator in the second case. The question is how much more there can be.