does every adjoint orbit of a Lie group go through the Cartan subalgebra?

Question

A naive question from a physicist, so forgive the lack of rigor. Consider a Lie group, acting on its Lie algebra by the adjoint action. Does every orbit go through the Cartan subalgebra? Alternatively, for which Lie groups is this true? (I view the statement as a generalization of the elementary fact that every Hermitian matrix can be diagonalized by a unitary transformation.) It would be nice if it held at least for real, compact, and semi-simple Lie algebras. Thank you for any hint or reference to literature!

Answer 1

If G is a compact and connected Lie Group,that must be true!!About this idea,you can read this book(Differential Geometry,Lie Groups and Symmetric Spaces,author:Helgason) Chapter V,6th section.