$ SO_n $ is maximal among the proper connected closed subgroups of $ SU_n $ (at least this is true for small $ n $). What is a conceptual proof of this fact? Equivalently what is a proof that $ \mathfrak{so}_n $ is maximal in $ \mathfrak{su}_n $?
Can we perhaps make a general argument that for any irreducible symmetric space $ G/K $ it is the case that $ K $ is a maximal closed connected subgroup of $ G $?