  We use the Fradkin-Vasiliev procedure to construct the full set of
non-abelian cubic vertices for totally symmetric higher spin gauge fields in
anti-de Sitter space. The number of such vertices is given by a certain
tensor-product multiplicity. We discuss the one-to-one relation between our
result and the list of non-abelian gauge deformations in flat space obtained
elsewhere via the cohomological approach. We comment about the uniqueness of
Vasiliev's simplest higher-spin algebra in relation with the (non)associativity
properties of the gauge algebras that we classified. The gravitational
interactions for (partially)-massless (mixed)-symmetry fields are also
discussed. We also argue that those mixed-symmetry and/or partially-massless
fields that are described by one-form connections within the frame-like
approach can have nonabelian interactions among themselves and again the number
of nonabelian vertices should be given by tensor product multiplicities.
