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what does it mean that all Cartan subalgebras are conjugate under automorphisms of the Lie algebra if the field is algebraically closed?

badan
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    What don't you understand about that statement? – Qiaochu Yuan Apr 04 '15 at 02:47
  • @QiaochuYuan if $\psi \in Aut (L)$ and $H_1$ and $H_2$ are Cartan subalgebras, does it mean $\psi^{-1} H_1 \psi = H_2$? for me this doesn't make any sense! – badan Apr 04 '15 at 02:58
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    It means $\psi(H_1) = H_2$. The term "conjugate," I guess, makes the most sense in the special case that $\psi$ comes from the adjoint action of a Lie group $G$ on its Lie algebra $\mathfrak{g}$. E.g. in the case $G = \text{GL}_n, \mathfrak{g} = \mathfrak{gl}_n$ you are literally conjugating by some invertible matrix. – Qiaochu Yuan Apr 04 '15 at 03:13

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