In

Rothschild, Linda Preiss; Stein, Elias M.,Hypoelliptic differential operators and nilpotent groups, Acta Math. 137(1976), 247-320 (1977). ZBL0346.35030. PDF at archive.ymsc.tsinghua.edu.cn

they mentioned the free nilpotent Lie algebra $\mathfrak{R}_{F,s}$ in Part I, section 3, example 4. And they introduced $n_s$ and $m_s$ in Part II, section 7, where

- $n_s$ is the dimension of the free nilpotent Lie algebra $\mathfrak{R}_{F,s}$ of step $s$ on $n$ generators,
- $m_s$ is the dimension of the linear space spanned by all commutators of the vector fields $\{W_k\}_{k=1}^n$ ($n$ is the number of vector fields) of lengths $\leq s$ restricted to $\xi$.

I can understand $m_s$ easily but It's difficult to understand $n_s$ to me. I think I need some example for $n_s$. Consider the vector fields: $$X=(X_1,X_2)=(\partial_1,x_1\partial_2)$$ Its $m_1=1$ restricted to point $(0,x_2)$ and $m_1=2$ at others, while $m_2=2$ at any point in $\mathbb{R}^2$. What is $n_1$ and $n_2$ for this example? why?

And by the theorem 4 and introduction in this paper, if we lift the vector fields as $$\widetilde{X}=(\widetilde{X_1},\widetilde{X_2})=(\partial_1,x_1\partial_2+\partial_3)$$ then, $m_2=3$ and $m_2=n_2$. Why does $m_2=n_2$?

Thanks for your help!