Suppose I have an action of a linear reductive group ($GL(2,\mathbb{C})^2$ in this case) on a complex vector space (of dimension $16$) and I want to compute explicitly the ring of invariants of this representation. I already have the candidates for the generators (certain polynomials of $16$ variables) but I need to find the relations between them.

My question is the following - is it possible to force Magma, Sage or other program to find these polynomial relations (hopefully the whole ideal of relations...) for me? I don't expect them to be complicated, but they are way too big to manage by hand. I know the topic is discussed in several books (Derksen & Kemper, Sturmfels etc.) but are there precise, already implemented algorithms doing this or should one do it by himself?

PS. I saw Algorithms in Invariant Theory topic, but that's not exactly what I'm asking about.