I have a very precise question. I'm stuck on a stupid thing, at page 35 of Stratified lie groups and potential theory for their sub-Laplacians by Bonfiglioli,Lanconelli e Uguzzoni. In Remark 1.3.7 they are dealing with a $\delta_\lambda$-homogeneous vector field $X=\sum_{i=1}^Na_i\partial_{x_i}$ on $\mathbb R^N$ of degree $n$ and they arrive to say that $a_i$ are $\delta_\lambda$-homogeneous functions of degree $\sigma_i-n$. Up to here everything seems clear. What I don't understand, and it is important for my purposes, is the conclusion they make: if $n>0$ this means that $$a_i=a_i(x_1,\dots,x_{i-1}),$$ that is, $a_i$ does not depend on $x_i,\dots,x_N$.
Why?
In the unlikely event you're still looking for answers: if $a_i$ is $\delta_\lambda$-homogeneous of degree $\sigma_i - n$ then by equation (1.51) we know $a_i$ is a polynomial of the form $\sum_{|\alpha|_\sigma = \sigma_i - n} a_\alpha x^\alpha$. If $a_i$ depends on $x_j$ for any $j \geq i$, in which case $\alpha_j \geq 1$, then $|\alpha|_\sigma \geq \alpha_j\sigma_j \geq \sigma_j \geq \sigma_i > \sigma_i - n$, a contradiction.