Does triality survive in product Lie groups?

Question

Look at the following diagrams of Lie groups

        o                  o                         o
       /                                        
      /                                          
o----o    SO(8)    o----o     SU(3)xSO(4)    o====o     SO(5)xSO(4)
      \
       \
        o                  o                         o    

Note that they are not subgroups of SO(8), but still it could be said that their roots remember SO(8) in some ways. For instance, they are also isometry groups of seven dimensional manifolds. My particular question is, what does happen with triality of representations of SO(8)? Is there some similar simmetry, or remmants of it, in the other groups?