The Lie algebra generated by $e_1, e_2$ is spanned by the iterated Lie brackets of $e_1$ and $e_2$ ($e_1, e_2, [e_1, e_2], [e_1, [e_1, e_2]], [e_2, [e_1, e_2]], \ldots$). Is it also spanned by just $e_1, e_2$ and $[e_1, e_2]$ ?
The reason I ask is that in this Wikipedia article the parabolic Hörmander condition is stated just with $n+1$ iterations of the Lie bracket when considering $n+1$ vector fields and in other sources it is stated with an arbitrary number of iterations of the Lie bracket.
I would like to show that a particular SDE $dx = A_0(x)dt + A_1(x) dB_t$ does not satisfy the parabolic Hörmander condition and I was wondering if it would be enough to show that $A_1(x_0)$ and $[A_0, A_1](x_0)$ do not span the tangent space at a particular point $x_0$ like the Wikipedia article suggests.
No, a Lie algebra generated by $e_1,e_2$ need not be spanned by $e_1,e_2,[e_1,e_2]$. For example, consider the free-nilpotent Lie algebra $L_{2,3}$ generated by $e_1,e_2$ and Lie brackets $$ [e_1,e_2]=e_3,\; [e_1,[e_1,e_2]]=[e_1,e_3]=e_4,\; [e_2,[e_1,e_2]]=[e_2,e_3]=e_5 $$ It is $5$-dimensional, so that it cannot be spanned by three vectors alone.