Remark 2

Let $\rho$ be the adjoint representation of $\mathfrak{g}$ on $U(\mathfrak{g})$. Let $\alpha , x, n, h$ be as in (ii) above. We have $\rho (x^{(n)}).\mathscr{U}\subset \mathscr{U}$ by Lemma 2. On the other hand, if $p\in \mathbf{N}$,

$((h))_{(n)}($ ad $h)_{(n)}(n\alpha (h))_{(n)}$

$\rho x$ = $x$ = $x$

(no. 5, formula (13)), so $\rho ((^h_p)).\mathscr{U}\subset \mathscr{U}$. This proves that $\mathscr{U}$ is an admissible lattice in $U(\mathfrak{g})$, and it follows that $\mathscr{G}$ is an admissible lattice in $\mathfrak{g}$ (for the adjoint representation).