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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 $G_2$.

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    Elementary questions such as this one belong on MathStackExchange, not here. – Robert Bryant May 26 '15 at 10:33
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    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 May 26 '15 at 12:42
  • @YCor, Thanks for the answer. Would you please introduce me a reference for this statement and its proof? –  May 26 '15 at 14:10
  • 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 May 26 '15 at 15:03
  • @IgorRivin , Thanks. I see, very nice proof. –  May 26 '15 at 15:26
  • I'm voting to close this question as off-topic because answered in the comments. – abatkai May 26 '15 at 16:27
  • This is a valid question - many questions have fairly complete answers in comments. – Mark Bennet May 28 '15 at 20:41

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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

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