Are there any general results on when conjugate representations of a real Lie algebra are equivalent? I'm inclined to say that they are often not, but this is merely going on my case by case experience.
In particular if we know that the fundamental and antifundamental representations of the Lie algebra are inequivalent, can we deduce that all conjugate representations are? I feel that this should be possible, but can't get started with a proof.
Has anyone got either (a) some hints to get me started or (b) a good reference which might be able guide me through such a problem?
I genuinely don't know at present whether this is trivial or difficult, so any advice would be much appreciated.