What is the automorphism group of Lie group $E_6$

Question

I found out that there are outer automorphisms of compact Lie group $E_6$. For example complex conjugation $\tau$ as defined in this Yokota paper is outer automorphism of $E_6$ which fix $F_4$ subgroup. This means that automorphism group of $E_6$ is bigger than $E_6$ (embedded into $Aut(E_6)$ as inner automorphisms divided by 3-element center $\mathbb Z_3$). Is it known what is the full automorphism group of $E_6$ ?

Answer 1

For a compact simply-connected semisimple Lie group $G$ the quotient group $Out(G)=Aut(G)/Inn(G)$ is isomorphic to the automorphism group of the Dynkin diagram, which is $\mathbb{Z}/2$ in case of $E_6$, see here. Now we have the exact sequence $$ 1\rightarrow Inn(G) \rightarrow Aut(G) \rightarrow Out(G)\rightarrow 1, $$ which is an extension problem. For the automorphism groups of the corresponding semisimple Lie algebras, the exact sequence splits, so that $Aut(\mathfrak{g})$ is a semidirect product of $Inn(\mathfrak{g})$ and $Out(\mathfrak{g})$.