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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

wood
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    See this question. – Dietrich Burde Dec 05 '21 at 09:21
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    Crack open Humphreys' Introduction to Lie algebras. He proves this in Chapter 4 Section 16, I believe, for any Lie algebra over an algebraically closed field. There's probably a more straightforward approach focusing on semisimple Lie algebras available elsewhere. – Callum Dec 05 '21 at 14:43
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    That's a non-trivial theorem that should be covered in every good source on the theory of semisimple Lie algebras (as you rightly say, it guarantees that the root system theory is independent from the choice of a CSA). On this site I found https://math.stackexchange.com/q/2664495/96384 and https://math.stackexchange.com/q/3157402/96384. Googling just gave me https://projecteuclid.org/journals/hiroshima-mathematical-journal/volume-32/issue-2/On-the-conjugacy-theorem-of-Cartain-subalgebras/10.32917/hmj/1151007553.full – Torsten Schoeneberg Dec 05 '21 at 21:03
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    By the way, over an arbitrary characteristic $0$ field $k$, conjugacy of maximal $k$-split toral subalgebras in semisimple Lie algebras is shown by purely algebraic (but technically challenging) methods in G. Seligman, Rational Methods in Lie Algebras I.3. I gave a 2-page sketch of that proof in Thm 3.1.16 of my thesis. This implies the case here. The analogous assertion in the reductive group case, that all $k$-split tori are conjugate, is Theorem 4.21 in the first part of Borel-Tits' Groupes reductifs, http://www.numdam.org/item/PMIHES_1965__27__55_0.pdf – Torsten Schoeneberg Dec 06 '21 at 04:49

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