The space $C^{\infty}(\mathbb{R}^d)$ is a Fréchet space with seminorms $$ \max_{\alpha \in \mathbb{N}^d, |\alpha| \leq k} \sup_{x \in K} |D^{\alpha}f(x)| $$ where $K$ is compact and $k \in \mathbb{N}$. Here I use, for any $\alpha = (a_1,\ldots,a_d)$, the notation $|\alpha| = a_1 + \ldots + a_d$ and more importantly: $$ D^{\alpha}f = \left(\frac{\partial}{\partial x_1}\right)^{a_1} \cdots \left(\frac{\partial}{\partial x_d}\right)^{a_d} f $$
I am wondering if instead of using $\displaystyle\frac{\partial}{\partial x_i}$, one could use any family of smooth vector fields spanning the tangent space at any point. For example when $d=2$: $$ (1+y^2)\frac{\partial}{\partial x} \qquad \frac{\partial}{\partial y} $$ In particular, I am interested in how these seminorms are related to the usual ones and what happens when the operators do not commute. Also, is the assumption that they span the tangent space at any point needed? Here is an example when the family is always a basis, except at the origin: $$ y\frac{\partial}{\partial x} \qquad \frac{\partial}{\partial y} $$