J$. Then $U$ is open in $\RR^{2g}$. On the other hand, the set $$S := \{(\sigma_1(\eta),\dots, \sigma_g(\eta), \sigma_1(\beta), \dots, \sigma_g(\beta)) : (\eta, \beta) \in K^{\oplus 2} \}$$ is dense in $\RR^{2g}$. (This follows for example from [@Mar p. 135].) We conclude that the intersection of $S$ with $U$ is nonempty.

Considering the definitions of $U$ and $S$, we conclude that (after scaling by a positive integer if necessary) there exist $\eta, \beta \in {{\mathcal O}}_K$ with the property that

1.  [$\sigma_k(\eta) < 0$ if $k\in I$ and $\sigma_k(\eta) > 0$ if $k \not \in I$;]{}

2.  [$\sigma_k(\beta) < 0$ if $k\in J$ and $\sigma_k(\beta) > 0$ if $k \not \in J$;]{}

3.  [ $\sigma_k(\eta) + \sigma_k(\beta) < 0$ if $k \in I \cup J$.]{}

Consequently, the Hermitian forms $H_\eta$, $H_\beta$, and $H_\eta + H_\beta = H_{\eta + \beta}$ have, respectively, exactly $p$, $q$, and $p+q$ negative eigenvalues.

Let $\chi_{H_\eta}$ and $\chi_{H_\beta}$ be semi-characters for $\Lambda_{{\mathbf{z}}}$ with respect to $H_\eta$ and $H_\beta$ respectively. The line bundles $L(H_\eta, \chi_{H_\eta})$, $L(H_\beta, \chi_{H_\beta})$, $L(H_\eta +H_\beta, \chi_{H_\eta} \chi_{H_\beta})$ are non-degenerate, see [@Mum p. 80], and satisfy the relation $${\operatorname{i}}(L(H_\eta, \chi_{H_\eta})) + {\operatorname{i}}(L(H_\beta, \chi_{H_\beta})) = {\operatorname{i}}(L(H_\eta +H_\beta, \chi_{H_\eta} \chi_{H_\beta}))$$ by the corollary on p. 151 of 