Take $L$ a semisimple complex Lie algebra and $H$ a maximal toral subalgebra. I want to prove that given a second maximal toral subalgebra $H'$ then exists an inner automorphism of $L$, $\omega$, such that $\omega(H)=H'$.
I think that a such automorphism must exist from the unicity of the root system but I can't prove that it can be taken inner.
Thanks in advance