On the trivectors of a 6-dimensional symplectic vector space. IV

Let V be a 6-dimensional vector space over a field F equipped with a nondegenerate alternating bilinear form f. The group GL(V) has a natural action on the third exterior power ⋀3V of V which defines five families of nonzero trivectors of V. Four of these families are always orbits regardless of the structure of the underlying field F. The orbits contained in the fifth family are in one-to-one correspondence with the quadratic extensions of F that are contained in a fixed algebraic closure of F. We will divide those orbits corresponding to the nonseparable quadratic extensions into suborbits for the action of the symplectic group Sp(V,f)≅Sp6(F) associated with (V,f) on ⋀3V.