I know that $(z_1,z_2)$ are the affine\inhomogeneous coordinates on the complex projective space $CP^2$. Now I have four one forms $(Y_1,Y_2, Y_3, Y_4)$. I want to rewrite the Fubini Study metric on $CP^2$ in terms of the forms $(Y_1,Y_2, Y_3, Y_4)$ .  The expressions for the one forms in terms of $(z1,z2)$ are given below
\begin{split}
&&Y1=\frac{|z_1|^2+|z_2|^2\sqrt{1+|z_1|^2+|z_2|^2}}{(|z_1|^2+|z_2|^2)(1+|z_1|^2+|z_2|^2)}dz1-\frac{z1\bar{z}_2(-1+\sqrt{1+|z_1|^2+|z_2|^2})}{(|z_1|^2+|z_2|^2)(1+|z_1|^2+|z_2|^2)}dz2\\
&&Y2=\frac{|z_2|^2+|z_1|^2\sqrt{1+|z_1|^2+|z_2|^2}}{(|z_1|^2+|z_2|^2)(1+|z_1|^2+|z_2|^2)}dz2-\frac{z2\bar{z}_1(-1+\sqrt{1+|z_1|^2+|z_2|^2})}{(|z_1|^2+|z_2|^2)(1+|z_1|^2+|z_2|^2)}dz1\\
&&Y_3=-\bar{Y}_1\\ 
&&Y_4=-\bar{Y}_2
\end{split}

I have been able to workout the following relations
$$
\begin{split}
\frac{\bar{z}_1 dz_1+\bar{z}_2 dz_2}{1+|z_1|^2+|z_2|^2} &=\bar{z}_1Y_1+\bar{z}_2Y_2\\
\frac{z_1d\bar{z}_1+z_2 d\bar{z}_2}{1+|z_1|^2+|z_2|^2}&=-(z_1Y_3+z_2Y_4)
\end{split}
$$
 Is there a way to get the relation using the above condition so that I can rewrite the Fubini-Study metric in terms of these forms?
Special Case: To make my question clear let us consider the case $z2=0$. Then we see the relationship reduces to the following
$$
\begin{split}
\frac{ dz_1}{1+|z_1|^2} &=Y_1\\
\frac{d\bar{z}_1}{1+|z_1|^2}&=-Y_3
\end{split}
$$
Taking the product of the above equation gives
$$
\begin{split}
ds^2=\frac{ dz_1d\bar{z}_1}{(1+|z_1|^2)^2}=-Y_1Y_3=Y_1\bar{Y}_1
\end{split}
$$
This is the Fubini Study metric on $CP^1$ with inhomogeneous/affine coordinates $(z_1,z_2)$. I am expecting to rewrite using the forms the Fubini-Study metric on $CP^2$.