I'm reading McMullen's paper "Billiards and Teichmüller Curves on Hilbert Modular Surfaces." I am stuck on understanding isomorphism (6.2) in the "Hilbert modular varieties" subsection of section 6 on page 18.
Here is the set-up: Let $K = \mathbb{Q}[\alpha]$ be a totally real number field over $\mathbb{Q}$ of degree $g$, where $K/\mathbb{Q}$ is a Galois extension. Consider the lattice $L_0 = \mathbb{Z}^{2g}$ with the standard symplectic form $E_0$ satisfying $E_0(a_i, b_j) = \delta_{ij}$. Suppose we have an embedding $K \to \text{End}(L_0) \otimes \mathbb{Q}$ sending the generator $\alpha$ to an endomorphism $T$ that satisfies $E_0(Tx, y) = E_0(x, Ty)$.
McMullen says that because $T$ is self-adjoint (in the above sense), we can find a $K$-linear isomorphism $$ L_0 \otimes \mathbb{Q} \cong K^2 $$ that sends $E_0$ to the "standard symplectic form" $E_1$ on $K^2$ $$ E_1(k, \ell) = \text{Tr}_\mathbb{Q}^K \det \begin{pmatrix} k_1 & k_2 \\ \ell_1 & \ell_2\end{pmatrix}. $$ The trace here is the field trace, i.e., the sum over the Galois conjugates.
Does anyone see how to use the correspondence between $\alpha$ and $T$ to build the isomorphism carrying one symplectic form to the other? Moreover, does anyone have a reference discussing this symplectic form on $K^2$?