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We determine convex PBW-type Lyndon bases for two-parameter quantum groups$U_{r,s}(F_4)$ with detailed commutation relations. We construct afinite-dimensional Hopf algebra $\mathfrak u_{r,s}(F_4)$, as a quotient of$U_{r,s}(F_4)$ by a Hopf ideal generated by certain central elements, which ispointed, and of a Drinfel'd double structure under a certain condition. All ofHopf isomorphisms of $\mathfrak u_{r,s}(F_4)$ are determined which areimportant for seeking the possible new pointed objects in low order with$(\ell, 210)\ne 1$. Finally, necessary and sufficient conditions for $\mathfraku_{r,s}(F_4)$ to be a ribbon Hopf algebra are singled out by describing theleft and right integrals.

Convex PBW-type Lyndon Bases and Restricted Two-Parameter Quantum Group of Type $F_4$