  Majid developed in \cite{majid3} his double-bosonization theory to construct
$U_q(\mathfrak g)$ and expected to generate inductively not just a line but a
tree of quantum groups starting from a node. In this paper, the authors confirm
the Majid's first expectation (see p. 178 \cite{majid3}) through giving and
verifying the full details of the inductive constructions of $U_q(\mathfrak g)$
for the classical types, i.e., the $ABCD$ series. Some examples in low ranks
are given to elucidate that any quantum group of classical type can be
constructed from the node corresponding to $U_{q}(\mathfrak{sl}_2)$.
