Proposition 47

Let G be a Lie group, H and E Lie subgroups of G and suppose that hEh$^{-1}$ = E for all h \in H. Then $\mathscr{T}^{(\infty)}(H) \subset \mathscr{T}^{(\infty)}(E) \subset \mathscr{T}^{(\infty)}(E)$. In particular, $\mathrm{Ad}(H)(L(E)) \subset L(E)$ and $[L(H), L(E)] \subset L(E)$.

If $t \in \mathscr{T}^{(\infty)}(H)$ and $t' \in \mathscr{T}^{(\infty)}(E)$, then $t \otimes t' \in \mathscr{T}^{(\infty)}(H \times E)$ and the image of $H \times E$ under the mapping $(g, g') \mapsto gg'g^{-1}$ is contained in E.