Where does $[\delta_{\xi_1},\delta_{\xi_2}]=\delta_{(\xi_2\partial \xi_1-\xi_1\partial\xi_2)}$ come from in CFT?

Question

The authors in the book "Bacis concepts of string theory" say on page 69 that

"Using the commutation properties of infinitesimal conformal transformations $$[\delta_{\xi_1},\delta_{\xi_2}]=\delta_{(\xi_2\partial \xi_1-\xi_1\partial\xi_2)},\tag{4.28}$$ we find $$T(z)T(w)=\frac{c/2}{(z-w)^4}+\frac{2T(w)}{(z-w)^2}+\frac{\partial T(w)}{(z-w)}+\text{finite terms.}\tag{4.29}$$

I would really grateful if someone could tell me where does the expression $(4.28)$ come from? and how can we find the expression $(4.29)$ from $(4.28)$?

The authors have mentioned that $$\delta_{\xi}\phi(w)=-[T_{\xi},\phi (w)]\tag{4.20}$$ on page 68.

Answer 1

(4.28) can be obtained by direct application of (4.9): $$ \delta_{\xi_1} \delta_{\xi_2} \phi(z) = (- h \partial \xi_1 \partial \xi_2 + \xi_1 (\partial \xi_2) \partial + \xi_1 \xi_2 \partial \partial) \phi(z), $$ and then notice that the first and third terms are symmetric under $1 \leftrightarrow 2$, only the second term survives in commutator, which gives (4.28).

I am not sure if one can go from (4.28) to (4.29) directly. It has already been asked (Derivation of the $TT$ OPE?) however it gets no answears yet.