Consider the Weierstrass representation $$f(z)=\text{Re}\int ((1-g^2),i(1+g^2),2g)\omega$$, where $g(z)$ is a meromorphic function and $\omega(z)$ is a holomorphic 1 form.
I'm trying to derive the following expression of second fundamental form $$II=-\omega dg-\overline{ \omega dg}$$. This is (5.9) in the notes https://www.math.titech.ac.jp/~kotaro/class/2016/geom-f/lecture-05.pdf
Following the sketch of the proof in the note. We should have $$II=(f_{zz}\cdot\nu)dz^2+2(f_{z\bar{z}}\cdot\nu)dz\bar{dz}+(f_{\bar{z}\bar{z}}\cdot\nu)\bar{dz}^2$$
Since all components of $f_z$ are holomorphic so $f_{z\bar{z}}=0$ and hence the second term vanishes. The first and last terms are conjugate so it is enough to prove $f_{zz}\cdot\nu dz^2=-\omega dg$.
I have trouble with showing this. If we use the $\nu=\frac{1}{1+|g|^2}(2\text{Re}g, 2\text{Im}g,|g|^2-1)$ and compute directly it will be too tedious and I failed to reach the RHS of the equality. I wonder if this is the only way to derive the second fundamental form. Are there any easier method to do this ?