  We study the ring of Weyl invariant $E_8$ weak Jacobi forms. Wang recently
proved that the ring is not a polynomial algebra. We consider a polynomial
algebra which contains the ring as a subset and clarify the necessary and
sufficient condition for an element of the polynomial algebra to be a Weyl
invariant $E_8$ weak Jacobi form. This serves as a new algorithm for
constructing all the Jacobi forms of given weight and index. The algorithm is
purely algebraic and does not require Fourier expansion. Using this algorithm
we determine the generators of the free module of Weyl invariant $E_8$ weak
Jacobi forms of given index $m$ for $m\le 20$. We also determine the lowest
weight generators of the free module of index $m$ for $m\le 28$. Our results
support the lower bound conjecture of Sun and Wang and prove explicitly that
there exist generators of the ring of Weyl invariant $E_8$ weak Jacobi forms of
weight $-4m$ and index $m$ for all $12\le m \le 28$.
