In $\mathbb{C}^2$, I want to show that $\langle x,y \rangle =xAy^*$ is an inner product space. I almost done with other properties of inner product but left only one property: $\langle x,x \rangle >0$ if $x \neq 0$.

To prove this,
\begin{align*} \langle x,x \rangle&=(x_1, x_2)A(x_1, x_2)^*\\ &=\|x_1 \|^2-ix_2\overline{x_1}+ix_1\overline{x_2}+\|x_2 \|^2\\ &=\|x_1 \|^2 +2\mathrm{Re}(ix_1\overline{x_2})+\|x_2 \|^2 \end{align*}
where $\mathrm{Re}(z)$ is the real part of the complex number $z$.

I can't follow the last equality. Why does $\mathrm{Re}(z)$ appear here?