The simple Lie algebra $W(1,2)^{(2)}$ of dimension 3 over $GF(2)$ obviously contains 8 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.