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

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