Let $G$ be a lie group with lie algebra $\frak{g}$. Let $Aut(\frak g)$ be the automorphism group of $\frak{g}$.
Its clear to me that $Aut(\frak{g})$ $\subset GL(\frak{g})$ since any automorphism of $\frak{g}$ is also a linear transformation. Now i have several questions:
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How do i prove that $Aut(\frak{g})$ is actually a lie subgroup of $GL(\frak{g})$? I've tried playing with actions to realize $Aut(\frak g)$ as an orbit or an isotropy subgroup but nothing worked out…
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Say $H \subset G$ is an abstract subgroup (not necessarily lie). Are the following statements equivalent?
$$\{T_e H \text{ is closed under the lie bracket of } \frak{g} \}$$
$$ \{H \text{is a lie subgroup of } G\}$$
- How does the lie algebra of $Aut(\frak g)$ look like? When does $Aut(\frak g)$ $= G$? what does $Aut(\frak g)$ tell me about $G$?
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