\ ^t \left[
\begin{matrix} \sigma_1(\eta_1) &
                  \dotsc &
                  \sigma_1(\eta_{2 m}) &
                  \dotsc &
                  \sigma_e(\eta_1) &
                  \dotsc   &
                  \sigma_e(\eta_{2m}) \end{matrix} \right]$$ is a lattice [@BL §9.2].

A consequence of [@BL Proposition 9.2.3, p. 250], is that $$(X, H, \iota) \cong (\CC^g / J_Z(\mathcal{M}) , H_Z, \rho)$$ for some $Z = (Z_1 ,\dots, Z_e )$, $Z_i \in \mathfrak{h}_m^{\oplus e}$, some $\ZZ$-module $\mathcal{M} \subseteq K^{\oplus 2}$, of rank $2 m$ such that $${\operatorname{Tr}}_{K / \QQ} \left(\sum\limits_{ \ell = 1}^m a_\ell b_{m + \ell} \right) - {\operatorname{Tr}}_{K / \QQ}\left(\sum\limits_{\ell = 1}^m a_{m + \ell} b_\ell \right) \in \ZZ \text{,}$$ for all $\mathbf{a} = (a_1,\dots, a_{2 m})$, $\mathbf{b} = (b_1,\dots, b_{2 m}) \in \mathcal{M} \subseteq K^{\oplus 2m}$, and some positive definite Hermitian from $H_Z$ on $\CC^g$, whose imaginary part is integral on $J_Z(\mathcal{M})$.

Useful for our purposes is the fact that the Hermitian form $H_Z$ is defined by $$H_Z(\mathbf{x}, \mathbf{y}) :=  \ ^t \mathbf{x} {\operatorname{diag}}({\operatorname{Im}}Z_1 ,\dots, {\operatorname{Im}}Z_e)^{ - 1} \overline{\mathbf{y}}$$ for all $\mathbf{x}$, $\mathbf{y}$ $\in \CC^g$.

On the other hand if $\eta \in K$, then multiplying the matrix representation of $H_Z$ with respect to the standard basis of $\CC^g$ with $\rho(\eta)$ we see that every nonzero $\eta \in K$ determines a non-degenerate Hermitian form $H_{Z, 