Remark 3

Prop. 6 shows that the canonical homomorphism

$\iota : T_Q/$Im(T$_P$)$\rightarrow$ Aut$_0(\mathfrak{g})/$Aut$_e(\mathfrak{g})$,

induced by the diagram in no. 2, is surjective. In particular, Aut$_e(\mathfrak{g})$ contains the derived group of Aut$_0(\mathfrak{g})$; we shall see (§11, no. 2, Prop. 3) that they are actually equal. Moreover, it can be shown that $\iota$ is injective, in other words that $f(T_Q)\cap$ Aut$_e(\mathfrak{g}) =f'(T_P)$,

(cf. §7, Exerc. $26d$)$)$.