One considers on a domain \(\Omega\) in \(\mathbb{R}^ n\) a self-adjoint second order subelliptic operator with positive characteristic. It can be written \[ Lf= h^{-1} \sum_{i,j} \partial_ i a_{ij} \partial_ j\cdot hf \] with \(h\), \(h^{-1}\), \(a_{ij}\in C^ \infty\) and real, and with the matrix \((a_{ij})\) non-negative. One associates to \(L\) a distance \(d: d(x,y)\leq r\) means that there is an absolutely continuous part \(\ell(t)\) starking at \(x\) and ending at \(y\) such that the speed vector \(\xi=\dot\ell(t)\) is almost everywhere subunit to \(L\), i.e. \[ \bigl(\sum \xi_ i\lambda_ i\bigr)^ 2\leq \sum a_{ij} \lambda_ i \lambda_ j,\quad \lambda_ i\in\mathbb{R}. \] The operator \(L\) is said to be a Hörmander operator if it can be written \[ L=\sum X^*_ j X_ j, \] where \(X_ j\) are \(C^ \infty\) vector fields, and \(X^*_ j\) is the formal adjoint of \(X_ j\). Let \(L_ 1\), \(L_ 2\) be two such operators on \(\Omega\), and let \(d_ 1\), \(d_ 2\) denote the corresponding distances. Consider the following conditions: (i) There exists \(C>0\) such that \[ d^ \alpha_ 1(x,y)\leq C d^ \beta_ 2(x,y),\quad x,y\in\Omega. \] (ii) There exists \(C>0\) such that \[ \| L^ \beta_ 2 f\|_{L^ 2}\leq C\bigl(\| L^ \alpha_ 1 f\|_{L^ 2}+ \| f\|_{L^ 2}\bigr),\quad f\in C^ \infty_ 0(\Omega). \] One proves that if \(L_ 1\) and \(L_ 2\) are subelliptic Hörmander operators, and if \[ d_ 1(x,y)\leq C| x- y|^{1/2}, \] then (i) implies (ii) for arbitrary \(\alpha,\beta>0\). Conversely if (ii) holds for some fixed \(\alpha_ 0\), \(\beta_ 0>0\), then (i) holds for \(\alpha=\alpha_ 0\) and \(\beta=\beta_ 0- \varepsilon\) and arbitrary \(\varepsilon>0\). This is obtained as a consequence of a basic commutation result, and most of the paper is devoted to its proof. Let \(A\) be a Hörmander operator, and let \(S_ 1,\dots,S_ k\) be pseudo-differential operators, \(S_ j\in OPS^{n_ j}_{1,0}\), \(n_ j\in\mathbb{R}\). Then, for \(m\in\mathbb{R}\), \(\sigma\in\mathbb{C}\), \[ \|[\cdots[[A^ \sigma,S_ 1],S_ 2]\cdots S_ k]f\|_ m\leq\| A^{\sigma-{k\over 2}} f\|_{m+n_ 1+\cdots+n_ k}, \] where \(\| f\|_ \alpha\) is the standard Sobolev norm.