Corollary 5

Let G be a connected compact Lie group. Then Int(G) is the identity component of the Lie group Aut(G) (Chap. III, §10, no. 2).

Let $f\in$ Aut(G)$_0$. Then $f$ induces an automorphism $f_1$ of $C(G)_0$ and an automorphism $f_2$ of D(G), and we have $f_1\in$ Aut(C(G)$_0$)$_0, f_2\in$ Aut(D(G))$_0$. Since Aut(C(G)$_0$) is discrete (General Topology, Chap. VII, §2, no. 4, Prop. 5), we have $f_1=$ Id; since D(G) is semi-simple, by Chap. III, §10, no. 2, Cor. 2 of Th. 1 there exists an element $g$ of D(G) such that $f_2(x) =gxg^{-1}$ for all $x\in$ D(G). For all $x\in C(G)_0$, we have $gxg^{-1}=x$ = $f_1(x)$; since $G = C(G)_0.D(G)$, it follows that $gxg^{-1}=f(x)$ for all $x\in G$, so $f=$ Int $g$.