I want to know whether an automorphism group of a simple Lie algebra over $GF(2)$, acts transitively on nonzero elements of Lie algebra or not? How can I check this property?
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closed as offtopic by YCor, Dietrich Burde, Ricardo Andrade, Chris Godsil, Suvrit Apr 7 '14 at 1:29



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

