I've been studying lately Sub-Riemannian Geometry from the notes of Enrico LeDonne and I am stuck with something. In particular, I am studying Carnot groups seen as Carnot-Carathéodory Spaces endowed with the $d_{CC}$ metric ($d_{CC}(p,q)$ is the infimum of length among all horizontal/admissible curves that start at p and end at q).
At some part of the text, it is mentioned that the Carnot group $(G, D, || \cdot ||)$ with $\mathfrak{g} = V_1 \bigoplus ... \bigoplus V_s$ stratifiable endowed with the distribution defined by $D_q = (L_q)_* V_1$ and $||v|| = ||(L_q)^* v||$ for all $q \in G$ and $v \in D_q$ makes true that $d_{CC}$ is left-invariant. I could prove it using the definition of left-invariance of the metric. However, the author mentions that by left-invariance of the metric it is true that $d_{CC}(e,p) < \infty$ for all $p \in G \rightarrow d_{CC}(p,q) < \infty$ for all $p,q \in G$. I've tried to prove that affirmation and I get why it must be true, but I haven't been able to get a detailed explanation.
Finiteness of $d_{CC}$ follows from Lemma 1.40 in Folland, Stein "Hardy spaces on homogeneous groups". (I know this result from this book, however you can find it in other books)
Lemma 1.40 says that if $G$ is a Carnot group then exists $N \in \mathbb N$ and $\delta>0$ (which depends only on $G$) such that for any $x \in G$ exist $x_1, \dots, x_N \in \exp(V_1)$ and $|x_i| \le \delta |x|$ where $|\cdot|$ is a homogeneous norm (equivalent to $d_{CC}$).
You can define $$ d_{CC}(x, y) = \inf\bigg\{\int\limits_a^b \|\dot\gamma(t)\|\,dt \mid \gamma\colon [a,b] \to G, \text{$\gamma$ is horizontal path with end points $x$ and $y$} \bigg\}. $$ We say that $\gamma$ piecewise smooth path is horizontal if $\dot\gamma(t) \in V_1(\gamma(t))$ for almost every $t$.
From lemma 1.40 one can find a path with finite length that's why $d_{CC}(e,x) < \infty$ for all $x\in G$. And $d_{CC}(x,y) = d_{CC}(e, x^{-1}y) < \infty$ for all $x,y\in G$.