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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$\begingroup$ See also mathoverflow.net/questions/161104/homogeneous-algebras. $\endgroup$– Dietrich BurdeCommented Apr 6, 2014 at 16:05
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$\begingroup$ yes. it was clear. I am sorry for asking repetative question. $\endgroup$– user118746Commented Apr 6, 2014 at 20:13
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Surely not. For example some elements are $\mathrm{ad}$-nilpotent and others are $\mathrm{ad}$-diagonable.
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$\begingroup$ what do you mean by ad-diagonable? $\endgroup$ Commented Apr 6, 2014 at 15:00
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$\begingroup$ @user40491, an element $x$ in the Lie algebra is ad-diagonalizable if the map $y\in g\mapsto [x,y]\in g$ is diagonalizable. $\endgroup$ Commented Apr 6, 2014 at 20:21