Remark 2

Let $\mathfrak{c}$ be the centre of $\mathfrak{g}$. Then, a lattice $\mathscr{H}$ in $\mathfrak{h}$ is permissible if and only if $Q(R^{\vee})\subset \mathscr{H}\subset P(R^{\vee})\oplus \mathfrak{c}$. The lattice $\mathscr{H}\cap \mathscr{D}\mathfrak{g}$ is then permissible in the Cartan subalgebra $\mathfrak{h}\cap \mathscr{D}\mathfrak{g}$ of $\mathscr{D}\mathfrak{g}$. There may exist permissible lattices $\mathscr{H}$ such that $\mathscr{H}\not= (\mathscr{H}\cap \mathscr{D}\mathfrak{g})\oplus (\mathscr{H}\cap \mathfrak{c})$ (cf. §13, no. 1.IX).