In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$ In questions 11) to 14), $G$ is an arbitrary closed subgroup of $\mathrm{GL}_n(\mathbf{R})$. $\mathbf{12}$ ▷ Let $A \in \mathcal{A}_G$ and $B \in \mathcal{A}_G$. Show that the application $$\begin{aligned} u : \mathbf{R} & \longrightarrow \mathcal{M}_n(\mathbf{R}) \\ t & \longmapsto u(t) = e^{tA} \cdot B \cdot e^{-tA} \end{aligned}$$ takes values in $\mathcal{A}_G$.
In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra:
$$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$
In questions 11) to 14), $G$ is an arbitrary closed subgroup of $\mathrm{GL}_n(\mathbf{R})$.
$\mathbf{12}$ ▷ Let $A \in \mathcal{A}_G$ and $B \in \mathcal{A}_G$. Show that the application
$$\begin{aligned} u : \mathbf{R} & \longrightarrow \mathcal{M}_n(\mathbf{R}) \\ t & \longmapsto u(t) = e^{tA} \cdot B \cdot e^{-tA} \end{aligned}$$
takes values in $\mathcal{A}_G$.