The question is: How do we define the Lie algebra generated by a set of vector fields?
I'm reading Kobayashi's book Transformation Groups in Differential Geometry and in the proof of the next theorem he uses that on the first line of the proof.
So my question is how do we define $g^*$? My first guess is that is the vector subspace spaned by $S$ together with the Lie bracket, but down the proof he proves that $S$ actually spans $g^*$ as vector spaces.