Compact and connected exceptional Lie group $G_2$

Question

The Lie group $G_2$ can be defined as the group of automorphisms of the Octonions $\Bbb Q_8$. It is often mentioned that it's compact and connected. But I am unable to find a proof that shows the compact and connectedness of $G_2$. Any help will be appreciated!

Answer 1

Last Friday I was just looking for references on exceptional Lie groups. One was the paper by I. Yokota, Exceptional Lie groups, found in arXiv. 0902.0431, where Theorem 1.9.3 in page 18 explicitly states the compactness and connectedness of $G_2$. The treatment is elementary and concrete. I like it.

Answer 2

The mentioned paper by Yokota gives two short proofs. In Theorem $1.2.2$ compactness is shown from the fact that $G_2$ (defined as automorphism group of the Cayley algebra) is a closed subgroup of the compact Lie group $O(8)$. Hence $G_2$ is compact itself. Furthermore, in Theorem $1.9.2$ it is shown that $$ G_2/SU(3)\cong S^6. $$ But since we already know that $SU(3)$ and $S^6$ are simply connected, the same follows for $G_2$.