I recently came across the concept of Cartan subgroup while studying physics. I took the definition from Wikipedia that a Cartan subgroup is a Lie sub-group whose Lie algebra is a Cartan subalgebra. We know Cartan subalgebras exist for finite-dimensional complex Lie algebras and are isomorphic to each other by an inner automorphism.
Is it true then that every Lie group has a Cartan subgroup? It would be extremely helpful if someone could confirm or deny the statement with proof or counter-example given. Reference to textbooks is also greatly appreciated, as I could not find any discussion of Cartan subgroup within the only book I have on Lie algebra at hand (GTM222 that is).
