The notion of a reductive dual pair makes sense over any field F, which we assume to be fixed throughout.
A reductive dual pair is called reducible if it can be obtained in this fashion from smaller groups, and irreducible otherwise.
Several classes of reductive dual pairs had appeared earlier in the work of André Weil.
Roger Howe proved a classification theorem, which states that in the irreducible case, those pairs exhaust all possibilities.
Moreover, if G = O(U) and G′ = Sp(V) are the isometry groups of U and V, then they act on W in a natural way, these actions are symplectic, and (G, G′) is an irreducible reductive dual pair of type I.