  Associated to each subset $J$ of the nodes $I$ of a Dynkin diagram is a
triangular decomposition of the corresponding Lie algebra $\mathfrak{g}$ into
three subalgebras $\widetilde{\mathfrak{g}_{J}}$ (generated by $e_{j}$, $f_{j}$
for $j\in J$ and $h_{i}$ for $i\in I$), $\mathfrak{n}^{-}_{D}$ (generated by
$f_{d}$, $d\in D=I\setminus J$) and its dual $\mathfrak{n}_{D}^{+}$.
  We demonstrate a quantum counterpart, generalising work of Majid and Rosso,
by exhibiting analogous triangular decompositions of $U_{q}(\mathfrak{g})$ and
identifying a graded braided Hopf algebra that quantizes
$\mathfrak{n}_{D}^{-}$. This algebra has many similar properties to
$U_{q}^{-}(\mathfrak{g})$, in many cases being a Nichols algebra and therefore
completely determined by its associated braiding.
