I am working with $G$ structures in differential geometry.
The definition I consider is the following a G-structure is a subbundle of the principal frame bundle such that the transition functions are ina subgroup $G$ of $GL(d)$.
Now, I know for example that for an almost complex structure (GL(d/2,C)) and a pre-symplectic (SP(d,R)) structure how to define compatibility by using invariant tensors $\omega \wedge \Omega = 0$ and the structure group is the intersection of the two groups, i.e. $U(d/2)$.
My question is: do there exist a general definition of compatibility of two structures, such that the structure group is the interection of the two groups defining the structures?