Can we write Lorentz transformations and dilations in terms of translations and special conformal transformations?
In V. Kac's book "Vertex algebra for beginners" 2nd edition, on p.7, Kac writes that "The group generated by the translations and the special conformal transformations is called the conformal group".
However, usually one says, not necessarily totally rigorously, that the conformal group is generated by translations, inversions, Lorentz transformations, dilations and special conformal transformations. That this is not optimal is shown in every book on CFT by showing that a translation conjugated by inversion gives a special conformal transformation, but I cannot prove Kac's statement. Since I was not able to find this statement anywhere else, now I wonder if it is true at all.
Definitions:
given $ x $ in $d$ dimensional Minkowski space $\mathbb{R}^{1,d-1}$, $c\in\mathbb{R}^d$, $\Lambda\in SO^+(1,d-1)$ (Lorentz group), $ \lambda\ne 0$, we call
$ x\mapsto x+c $ a translation,
$ x\mapsto \Lambda x$ a Lorentz transformation,
$ x\mapsto \lambda x$ a dilation,
$ x\mapsto \frac{x+|x|^2 b}{1+2\langle x , b\rangle + |x|^2|b|^2} $ a special conformal transformation.
So Kac's claim is that $\{transl,\, SCT\} = \{transl,\, Lorentz,\, dilat,\, SCT\}$.