# Action of automorphism group on Lie algebra [closed]

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, SuvritApr 7 '14 at 1:29

• This question does not appear to be about research level mathematics within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.

–  Dietrich Burde Apr 6 '14 at 16:05
yes. it was clear. I am sorry for asking repetative question. –  user40491 Apr 6 '14 at 20:13
@user40491, please consider accepting answers to questions you pose. –  Mariano Suárez-Alvarez Apr 6 '14 at 20:22

Surely not. For example some elements are $\mathrm{ad}$-nilpotent and others are $\mathrm{ad}$-diagonable.
@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 '14 at 20:21