Subgroups of the group $G_2 \times G_2$

Question

Does the group $G_2 \times G_2$ have the group $SO(7)$ (or its double cover $Spin(7)$) as its subgroup? Here, $G_2$ is the compact exceptional group $G_2$.

Answer 1

A product of groups $H_1\times H_2$ contains a simple group $S$ iff either either $H_1$ or $H_2$ contains $S$. Idem for Lie algebras. -- YCor

Consider the projection map $p_1$ onto the first factor. Either the map is an isomorphism restricted to $S$ (in which case $S$ is isomorphic to a subgroup of $H_1$) or it has a kernel. $S$ is simple, so the kernel must be all of $S.$ In which case, $S$ is contained in $H_2.$ -- Igor Rivin