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?



Action of automorphism group on Lie algebra
Surely not. For example some elements are $\mathrm{ad}$-nilpotent and others are $\mathrm{ad}$-diagonable.


