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I want to know whether an automorphism group of a simple Lie algebra over $GF(2)$, acts transitively on non-zero elements of Lie algebra or not? How can I check this property?

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closed as off-topic by YCor, Dietrich Burde, Ricardo Andrade, Chris Godsil, Suvrit Apr 7 at 1:29

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yes. it was clear. I am sorry for asking repetative question. –  user40491 Apr 6 at 20:13
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@user40491, please consider accepting answers to questions you pose. –  Mariano Suárez-Alvarez Apr 6 at 20:22
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1 Answer 1

Surely not. For example some elements are $\mathrm{ad}$-nilpotent and others are $\mathrm{ad}$-diagonable.

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what do you mean by ad-diagonable? –  user40491 Apr 6 at 15:00
    
@user40491, an element $x$ in the Lie algebra is ad-diagonalizable if the map $y\in g\mapsto [x,y]\in g$ is diagonalizable. –  Mariano Suárez-Alvarez Apr 6 at 20:21
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