Outer automorphisms of locally simple Lie algebras

The automorphism groups of locally simple Lie algebras over $\mathbb C$ are studied. The group of inner automorphisms of such algebras can be defined in a natural way, and it is normal in the automorphisms group. Hence a group of outer automorphisms of a locally simple Lie algebra can be defined. In contact to the finite-dimensional case, it is shown that the group of outer automorphisms is not necessarily finite. It is completely calculated in some special cases.