We assume that $|\raisebox{1.72218pt}{.}|$ is non-trivial and $\|\raisebox{1.72218pt}{.}\|=\|\raisebox{1.72218pt}{.}\|_{\mathscr{H}}$ for some finitely generated lattice $\mathscr{H}$ of $H$. Then a free basis $(e_{1},\ldots,e_{r})$ of $\mathscr{H}$ yields an orthonormal basis of $H$ with respect to $\|\raisebox{1.72218pt}{.}\|$ (cf. Proposition 1.14). Moreover, $\mathscr{H}=(H,\|\raisebox{1.72218pt}{.}\|)_{\leq 1}$.