The compact simple Lie groups $ SO_8(\mathbb{R}) $ and $ SO_9(\mathbb{R}) $ both have rank 4. The group $$ G=SU_3 \times SU_2 \times U_1 $$ also has rank 4. Does there exist a subgroup of $ SO_8(\mathbb{R}) $ or $ SO_9(\mathbb{R}) $ isomorphic to $ G $?
I already know that there is an inclusion of $ G $ into $ SU_5 $ given by putting $ SU_3 $ in the top 3 by 3 block, $ SU_2 $ in the bottom 2 by 2 block and then $ U_1 $ corresponds to the block scalar subgroup $ exp(-\frac{2 \pi i t}{3})I_3 \times exp(\frac{2 \pi i t }{2}) I_2 $ where $ I_n $ are identity matrices of the appropriate size and $ t $ parameterizes the subgroup. Since $ SU_5 $ naturally embeds in $ SO_{10}(\mathbb{R}) $ we can compose embeddings and get the an embedding of $ G $ into $ SO_{10}(\mathbb{R}) $.
