  We propose a new approach to study coideal algebras. It is well-known that
Manin triples (or equivalently Lie bi-algebra structures) are the requirement
to deform Lie algebras and to obtain quantum groups. In this paper, introducing
some particular automorphisms of Manin triples, we define new structures that
we call Lie bi-ideal structures. A link with coisotropic subalgebras is
explained. We show that their deformation provide coideal algebras. As
examples, we recover from our general construction the twisted Yangians, the
q-Onsager algebra and the augmented q-Onsager algebra. As an important
by-product, we find a new presentation for the twisted Yangians.
