Let $c=(c_1,c_2,c_3)$ be a complex vector. How can we see that $\|c\|^2=\|c\times \bar{c}\|$? Here the bar means component wise complex conjugation, the norm is the Hermitian norm, and the cross product is defined by Sarrus's rule (as far as I can tell). Would the equality be true if we use another norm?
If you need, assume that $c_i$ are holomorphic functions on a Riemann surface, satisfying the cyclicity condition. Then $x(z) = \Re\int_p^z c \, dz$ is a conformal minimal immersion, for any $p \in S$. Moreover, $x_u \times x_v = \frac{1}{2} c \times \bar{c}$. Since $x$ is conformal, $\|x_u \times x_v\| = \|x_u\|\cdot \|x_v\|$. I calculate $\|x_u \times x_v\|$ in many ways, but I fail to see that $\|x_u \times x_v\| = \frac{1}{2} \|c\|^2$. Obviously, $\|c\|^2 = \|x_u\|^2 + \|x_v\|^2 \not = 2 \|x_u\| \cdot \|x_v\|$.
The problem stems from the last formula in these notes.
Something is off here. Suppose for simplicity that all components of $c$ are real. If $c\times c$ is indeed a cross-product, then $\|c\times c\|=0\ne \|c\|^2$.