Subgroups of Exceptional Lie Groups

Question

I need to know if SO(13) is a subgroup of E_7. More generally, is there an easy way to tell if one Lie group is a subgroup of another simple Lie group?

Answer 1

https://ncatlab.org/nlab/show/maximal+compact+subgroup#ExamplesForLieGroups states that the maximal compact subgroup of E7 is $SU(8)/\mathbb{Z}_2$. It should be clear then that $SO(13)$ is a subgroup of E7 iff $SO(13)$ is a subgroup of $SU(8)/\mathbb{Z}_2$. This sounds very likely not the case but off the top of my head I'm not 100% certain how to show it.

Edit: $A_{13}$ is clearly a subgroup of $SO(13)$. Checking with https://en.wikipedia.org/wiki/Representation_theory_of_the_symmetric_group, it sounds like $A_{13}$ needs at least 12 dimensions to be faithfully represented (maybe 13, I'm not sure I'm reading it correctly -- but it doesn't matter) even in the complex numbers, so $A_{13}$ is not a subgroup of $SU(8)$. Thus $A_{13}$ is a subgroup of $SO(13)$ but not the maximal compact subgroup of E7, so E7 doesn't contain $SO(13)$.

I think that in general as long as one of your lie groups is compact, comparing with the maximal compact subgroup -- which is well understood -- is a good first step to establishing existence as a subgroup.