Let $A$ be a finite dimensional algebra over finite field (not necessarily associative). Then $A$ is said to be homogeneous if $Aut(A)$ acts transitively on the one-dimensional subspace of A. If A is homogeneous then either $A^2=0$ or $\text{dim}A=1$. Now I want to check this property for a finite dimensional simple Lie algebra over $GF(2)$. I want to know whether a simple lie algebra over $GF(2)$ is homogeneous or not?



homogeneous algebras
The simple Lie algebra $W(1,2)^{(2)}$ of dimension 3 over $GF(2)$ obviously contains 7 subspaces of dimension 1, but its automorphism group has order 6 (see e.g. section 5.3 of the paper "B. Eick: Some new simple Lie algebras in characteristic 2: J. Symbol. Comput. 45, 943 -- 951 (2010)"). It is then clear that this Lie algebra is not homogeneous.


