Multiple Cartan sub-algebras

Question

How is it that for a Semi-simple Lie Algebra there is not one Cartan Sub-Algebra?

I assume since there are multiple CSA's of a SS Lie algebra that must mean some of the ss elements of the Lie algebra get caught in other weight/root spaces corresponding to non-zero weights/roots? If not what happens to the SS elements not in the Cartan?

Thanks

Answer 1

All Cartan subalgebras of a given s.s. Lie algebra are conjugate under Lie algebra automorphism of the the large algebra anyway, so there is essentially only one choice. (At least this is true over fields of characteristic $0$, I don't know what how this plays out in prime characteristic.)