I know the definition of a Cartan subalgebra, but how do I actually find it explicitly for a particular Lie algebra over the complex numbers?

  • $\begingroup$ Do you mean for a Lie algebra? Are you working over the reals or the complex numbers? $\endgroup$ – Tobias Kildetoft Dec 3 '13 at 9:26
  • $\begingroup$ Sorry - yes I mean algebra of course. The example I have in mind is SU(2), so complex, but I'd like to know as generally as possible to find the subalgebra. $\endgroup$ – Alex Dec 3 '13 at 12:57
  • $\begingroup$ Hmm, I have not thought that much about how useful it is in practice, but if the Lie algebra is semisimple and we are working over the complex numbers, then Cartan subalgebras are the same as minimal Engel subalgebras which are again the same as maximal toral subalgebras. Also, Cartan subalgebras behave fairly well with respect to surjective morphisms (image of Cartan is Cartan, and a Cartan of the preimage of a Cartan is a Cartan of the original Lie algebra). $\endgroup$ – Tobias Kildetoft Dec 3 '13 at 13:23

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