The vector fields $\partial_x$, $x\partial_x$ and $x^2\partial_x$ in $\mathbb{R}$ have correspondant flows $x\mapsto x+t$, $x\mapsto e^t x$ and $x\mapsto (1-tx)^{-1}$ which are translation, dilations and inversions. By composing them, one easily checks that they generate the group $\mathrm{PGL}_2(\mathbb{R})$ of transformation of the form $$ x\mapsto \frac{ax+b}{cx+d} \quad\quad ad-bc\neq 0. $$
Can I compute a basis of the lie algebra of $\mathrm{PGL}_2(\mathbb{R})$ using these transformations?
I have been trying to define paths of transformations through the identity associated to these flows with the purpose to differentiate and see what happens when $t$ small, but I doing something wrong cause I do not obtain something with commutators similar to those of $\partial_x, x\partial_x$ and $x^2\partial_x$. In fact, I don't have it clear how to associate a path in $\mathrm{PGL}_2(\mathbb{R})$ to these transformations.