  The purpose of the paper is to build up the related theory of weakly
quasitriangular dual pairs suitably for non-standard $R$-matrices (irregular),
and establish the generalized double-bosonization construction theorem for
irregular $R$, which generalize Majid's results for regular $R$ in
\cite{majid1}. As an application, the type-crossing construction for the
exceptional quantum groups of types $F_{4}$, $G_{2}$ is obtained. This affirms
the Majid's expectation that the tree structure of nodes diagram associated
with quantum groups can be grown out of the node corresponding to
$U_q(\mathfrak{sl}_2)$ by double-bosonization procedures. Notably from a
representation perspective, we find an effective method to get the minimal
polynomials for the non-standard $R$-matrices involved.
