We establish a link between the quantum Lie algebras which appear in the work of Woronowicz in the context of differential calculi on quantum groups, and the braided Lie algebras introduced by Majid. We show that if T is the quantum tangent space (quantum Lie algebra) of a bicovariant first order differential calculus over a co-quasitriangular Hopf algebra (A,R), then a certain extension of it is a braided Lie algebra in the category of A-comodules. This is used to show that the quantum universal enveloping algebra U(T) is a bialgebra in the braided category of A-comodules. Surprisingly, we find out that this algebra is quadratic when the calculus is inner. Examples with this strange property include finite groups and quantum groups with their standard differential calculi.