Let $x, y \in I$ such that $\langle x, y \rangle = 0$. Since $\langle x, y \rangle \in \mathbb{R}$, applying
R2413: Complex conjugate of real number and the conjugate symmetry of inner products, we conclude
\begin{equation}
\langle y, x \rangle
= \overline{\langle x, y \rangle}
= \langle x, y \rangle
= 0
\end{equation}
$\square$
