Proposition 45

Suppose that G is finite-dimensional and that K is of characteristic 0. Let s be an integer $\geqslant 0$. Then the mapping $\pi : g \mapsto \mathrm{Ad}_{U_s(G)}(g)$ is an analytic linear representation of G on $U_s(G)$ and $L(\pi)a = \mathrm{ad}_{U_s(G)}a$ for all $a \in L(G)$.

The linear representation $\pi$ is a quotient of $\bigoplus_{r=0}^s \mathrm{Tr}(\mathrm{Ad})$ and is hence analytic. For $a \in L(G)$ and $x_1, x_2, \ldots, x_s$ in $L(G)$,