I am not really sure whether this topic belongs here or not, but since my example will be a certain one I will proceed here.... Someone please explain me what "isomorphism" means physically? For example what is the deal in saying that the proper orthochronous Lorentz group is isomorphic to SU(2)xSU(2)? I can understand as far that this example breaks everything in left-right handed movers, but I just can't generalize it in everything when we are talking about isomorphisms....
Maybe I should have stayed quiet. There is a fair bit going on, and I am not sure that I can explain all of it well. The statement in the OP is not true at the group level, but it is almost true at the level of Lie algebras (generators), i.e., it is true after the relevant real Lie algebras have been complexified. I might try a longer explanation later.