TITLE: Do all closed connected subgroups of $SO(2n+1)$ embed into $SO(2n)$?
QUESTION [8 upvotes]: Is every closed connected proper subgroup of $SO(2n+1)$ isomorphic (as a Lie group)
to a subgroup of $SO(2n)$? The answer is yes for abelian closed connected subgroups.

REPLY [10 votes]: As an answer to your weaker version:  No.  For example, $\mathrm{G}_2$ is a closed proper subgroup of $\mathrm{SO}(7)$, but it is not isomorphic to any subgroup of $\mathrm{SO}(6)$.  Of course, this follows from the fact that ${\frak{g}}_2$ has no nontrivial representation of dimension less than $7$.