  We develop two structure theorems for vector valued Siegel modular forms for
Igusa's subgroup \Gamma_2[2,4], the multiplier system induced by the theta
constants and the representation Sym^2. In the proof, we identify some of these
modular forms with rational tensors with easily handleable poles on P^3\C. It
follows that the observed modules of modular forms are generated by the
Rankin-Cohen brackets of the four theta series of the second kind.
