  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.
