  We prove the polynomiality of the bigraded ring $J_{*,*}^{w, W}(F_4)$ of weak
Jacobi forms for the root system $F_4$ which are invariant with respect to the
corresponding Weyl group. This work is a continuation of the joint article with
V.A. Gritsenko, where the structure of algebras of the weak Jacobi forms
related to the root systems of $D_n$ type for $2\leqslant n \leqslant 8$ was
studied.
