Corollary

If F is a finite-dimensional subspace of E, and if Φ is non-degenerate, the following conditions are equivalent:

a) F is non-isotropic. b) F^0 is non-isotropic. c) E is the direct sum of F and F^0.

Proposition 1 shows indeed that a) implies c), and that c) implies a) and b). Finally, if F^0 is non-isotropic, we have F ∩ F^0 ⊂ F^0 ∩ F^{00} = {0}; hence F is non-isotropic, which shows that b) implies a).