  Let K be an algebraically closed field of characteristic 0. It is well known
that any quasi-reductive Lie algebra is stable. However, there are stable Lie
algebras which are not quasi-reductive. This raises the question, if for some
particular class of non-reductive Lie algebras, there is equivalence between
stability and quasi-reductivity. In particular, it was conjectured by Panyushev
that these two notions are equivalent for biparabolic subalgebras of a
reductive Lie algebra. In this paper, we give a positive answer to this
conjecture.
