  We introduce the notion of a bicocycle double cross product (resp. sum) Lie
group (resp. Lie algebra), and a bicocycle double cross product bialgebra,
generalizing the unified products. On the level of Lie groups the construction
yields a Lie group on the product space of two pointed manifolds, none of which
being necessarily a subgroup. On the level of Lie algebras, similarly, a Lie
algebra is obtained on the direct sum of two vector spaces, none of which is
required to be a subalgebra. Finally, on the quantum level the theory presents
a bialgebra, on the tensor product of two (co)algebras that are not necessarily
sub-bialgebras, the semidual of which being a cocycle bicrossproduct bialgebra.
