In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$ If $A$ and $B$ are two matrices in $\mathcal{M}_n(\mathbf{K})$, their Lie bracket is defined by $[A, B] = AB - BA$. In questions 11) to 14), $G$ is an arbitrary closed subgroup of $\mathrm{GL}_n(\mathbf{R})$. $\mathbf{13}$ ▷ Deduce from question 12) that $\mathcal{A}_G$ is stable under the Lie bracket, i.e. $$\forall A \in \mathcal{A}_G, \forall B \in \mathcal{A}_G, [A, B] \in \mathcal{A}_G.$$
In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra:
$$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$
If $A$ and $B$ are two matrices in $\mathcal{M}_n(\mathbf{K})$, their Lie bracket is defined by $[A, B] = AB - BA$.
In questions 11) to 14), $G$ is an arbitrary closed subgroup of $\mathrm{GL}_n(\mathbf{R})$.
$\mathbf{13}$ ▷ Deduce from question 12) that $\mathcal{A}_G$ is stable under the Lie bracket, i.e.
$$\forall A \in \mathcal{A}_G, \forall B \in \mathcal{A}_G, [A, B] \in \mathcal{A}_G.$$