I am trying to prove that if $A \in M_n(\mathbb C)$ preserves norms then it also preserves inner products. I showed this for real matrices and I want to use this for this proof here. Let $f_n: \mathbb C^n \to \mathbb R^{2n}$ be the map $(x_1 + iy_2, x_2 + iy_2, \dots) \mapsto (x_1,x_2, \dots, y_1, y_2, \dots) $ and let $\rho_n: M_n(\mathbb C) \to M_{2n}(\mathbb R)$ be the homomorphism
$$ X + iY \mapsto \left ( \begin{array}{cc} X & Y \\ -Y & X \end{array}\right )$$
Let $A = X + iY$ preserve norms. Note that $R_A = f^{-1}_n \circ R_{\rho_n(A)}\circ f_n$ hence $f^{-1}_n \circ R_{\rho_n(A)}\circ f_n$ preserves norms. By the result for real matrices $f^{-1}_n \circ R_{\rho_n(A)}\circ f_n$ therefore also preserves inner products:
$$ \langle X, Y\rangle = \langle f_n^{-1}(R_{\rho_n(A)}(f_n(X)), f_n^{-1}(R_{\rho_n(A)}(f_n(Y))\rangle$$
And this is where I got stuck. How can I show that $ \langle X, Y\rangle = \langle R_A(X), R_A(Y)\rangle $ from here?