I have a problem:
We denote by $[X,\ Y]$ the commutator of $X$ and $Y$ defined by $$[X,\ Y]f(p)=X(Yf)(p)-Y(Xf)(p), \tag{1}$$ for any smooth function $f$ defined on a hypersurface $M$.
The form of the CR vector fields given in $(2)$: $$\Lambda = \frac{\partial}{\partial \bar{z}} -i\frac{\phi_{\bar{z}}}{1+i\phi_s} \frac{\partial}{\partial s}.\tag{2}$$ Show that
$$\begin{align*}[\Lambda,\ \bar{\Lambda}] &=i\dfrac{\phi_{z \overline{z}}}{1-i\phi_s}\dfrac{\partial }{\partial s}+i\dfrac{\phi_{\overline{z}{z}}}{1+i\phi_s}\dfrac{\partial }{\partial s}+i\phi_{z}\left ( \dfrac{1}{1-i\phi_s} \right )_{\overline{z}}\dfrac{\partial }{\partial s} \\ &\quad +i\phi_{\overline{z}}\left ( \dfrac{1}{1+i\phi_s} \right )_{{z}}\dfrac{\partial }{\partial s}-i\left (\dfrac{\phi_{\overline{z}}}{1+i\phi_s} \right )\left (\dfrac{\phi_{{z}}}{1-i\phi_s} \right )_{s}\dfrac{\partial }{\partial s} \\ &\quad +i\left ( \dfrac{\phi_{{z}}}{1-i\phi_s} \right )\left ( i\dfrac{\phi_{\overline{z}}}{1+i\phi_s} \right )_s\dfrac{\partial }{\partial s}. \end{align*}$$?????
with $\phi_{\bar z}=\dfrac{\partial \phi}{\partial \bar z},\ \phi_{s}=\dfrac{\partial \phi}{\partial s} $.
What I've tried:
I think that we need to use Wirtinger derivatives, so I write: $$\begin{align*}[\Lambda,\ \bar{\Lambda}]&=[\frac{\partial}{\partial \bar{z}} -i\frac{\phi_{\bar{z}}}{1+i\phi_s} \frac{\partial}{\partial s},\ \overline{\frac{\partial}{\partial \bar{z}} -i\frac{\phi_{\bar{z}}}{1+i\phi_s} \frac{\partial}{\partial s}}]\\ &=[\frac{\partial}{\partial \bar{z}} -i\frac{\phi_{\bar{z}}}{1+i\phi_s} \frac{\partial}{\partial s},\ \frac{\partial}{\partial {z}} +i\frac{\bar{\phi}_{z}}{1-i\bar{\phi}_{\bar s}} \frac{\partial}{\partial s} ] \end{align*}$$ Is it correct?
Any help will be appreciated! Thanks!