  The weak Jacobi forms of integral weight and integral index associated to an
even positive definite lattice form a bigraded algebra. In this paper we prove
a criterion for this type of algebra being free. As an application, we give an
automorphic proof of K. Wirthm\"{u}ller's theorem which asserts that the
bigraded algebra of weak Jacobi forms invariant under the Weyl group is a
polynomial algebra for any irreducible root system not of type $E_8$. This
approach is also applicable to $E_8$. Even if the algebra of $E_8$ Jacobi forms
is known to be non-free, we still derive a new structure result.

Number Theory;High Energy Physics - Theory