My understanding is that a left-invariant metric on a compact semi-simple lie group is defined in such a way that left translations are isometries. Bi-invariant metrics do not concern me for the moment. Are there isometries that are not left translations? Infinitesimally, are there killing vector fields that are not left-invariant?
In particular, how to find all isometries on the 3-sphere (i.e. SU(2)), for example? I can write three killing vector fields in terms of coordinate frame field of the ambient 4-dimensional euclidean space; but how to write them intrinsically on the sphere?