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

Double-bosonization and Majid's Conjecture, (II): cases of irregular $R$-matrices and type-crossings of $F_4$, $G_2$