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 $U_{ q}(\mathfrak{ sl}_2)$.
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