Why is this identity true?
$$\mathrm{Re}\langle v,iu\rangle=- \mathrm{Im}\langle u,v\rangle$$
If we start from LHS:
$$\mathrm{Re}\langle v,iu\rangle = \mathrm{Re}\left(-i\langle v,u\rangle\right)$$
How to proceed? I can't just take the $-i$ out of the $\mathrm{Re}$ function
Since $$ u=\Re u + i \Im u, \quad v = \Re v + i \Im v, $$ and $$ \langle v,u \rangle = v \overline{u}, $$ we have that $$ \Re \langle v,iu \rangle = \Re (-i \langle v,u \rangle) = \Re (-i v \overline{u})=\Re(-i (\Re v + i \Im v)(\Re u -i \Im u)). $$ Now expand and compare.
But, as suggested in the comments, there is a quicker answer. Call $z = \langle v,u\rangle$, and notice that multiplication by $-i$ is just a rotation of $-\pi/2$ in the complex plane. This rotation turns the real part into the imaginary part, and since $\langle u,v \rangle = \overline{z}$, you get the negative sign.