  We study the uniqueness of higher-spin algebras which are at the core of
higher-spin theories in AdS and of CFTs with exact higher-spin symmetry, i.e.
conserved tensors of rank greater than two. The Jacobi identity for the gauge
algebra is the simplest consistency test that appears at the quartic order for
a gauge theory. Similarly, the algebra of charges in a CFT must also obey the
Jacobi identity. These algebras are essentially the same. Solving the Jacobi
identity under some simplifying assumptions spelled out, we obtain that the
Eastwood-Vasiliev algebra is the unique solution for d=4 and d>6. In 5d there
is a one-parameter family of algebras that was known before. In particular, we
show that the introduction of a single higher-spin gauge field/current
automatically requires the infinite tower of higher-spin gauge fields/currents.
The result implies that from all the admissible non-Abelian cubic vertices in
AdS(d), that have been recently classified for totally symmetric higher-spin
gauge fields, only one vertex can pass the Jacobi consistency test. This cubic
vertex is associated with a gauge deformation that is the germ of the
Eastwood-Vasiliev's higher-spin algebra.
