  We give a necessary and sufficient condition for orbits of commutative
Hermann actions and actions of the direct product of two symmetric subgroups on
compact Lie groups to be biharmonic in terms of symmetric triad with
multiplicities. By this criterion, we determine all the proper biharmonic
submanifolds in irreducible symmetric spaces of compact type which are singular
orbits of commutative Hermann actions of cohomogeneity two. Also, in compact
simple Lie groups, we determine all the biharmonic hypersurfaces which are
regular orbits of actions of the direct product of two symmetric subgroups
which are associated to commutative Hermann actions of cohomogeneity one.
