  We show that if $g_\Gamma$ is the quantum tangent space (or quantum Lie
algebra in the sense of Woronowicz) of a bicovariant first order differential
calculus over a coquasitriangular 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 Woronowicz quantum universal enveloping algebra
$U(g_\Gamma)$ is a bialgebra in the braided category of $A$-comodules. We show
that this algebra is quadratic when the calculus is inner. Examples with this
unexpected property include finite groups and quantum groups with their
standard differential calculi. We also find a quantum Lie functor for
coquasitriangular Hopf algebras, which has properties analogous to the
classical one. This functor gives trivial results on standard quantum groups
$O_q(G)$, but reasonable ones on examples closer to the classical case, such as
the cotriangular Jordanian deformations. In addition, we show that split
braided Lie algebras define `generalised-Lie algebras' in a different sense of
deforming the adjoint representation. We construct these and their enveloping
algebras for $O_q(SL_n)$, recovering the Witten algebra for $n=2$.
