  Both in Majid's double-bosonization theory and in Rosso's quantum shuffle
theory, the rank-inductive and type-crossing construction for $U_q(\mathfrak
g)$'s is still a remaining open question. In this paper, working with Majid's
framework, based on our generalized double-bosonization Theorem proved in
\cite{HH2}, we further describe explicitly the type-crossing construction of
$U_q(\mathfrak g)$'s for $(BCD)_n$ series direct from type $A_{n-1}$ via adding
a pair of dual braided groups determined by a pair of $(R, R')$-matrices of
type $A$ derived from the respective suitably chosen representations. %which
generalize the lower rank cases constructed in \cite{HH1}. Combining with our
work in \cite{HH1,HH2,HH3}, this solves Majid's conjecture, that is, any
quantum group $U_q(\mathfrak g)$ associated to a simple Lie algebra $\mathfrak
g$ can be grown out of $U_q({\mathfrak {sl}}_2)$ inductively by a series of
suitably chosen double-bosonization procedures.
