A note on the geometry of linear Hamiltonian systems of signature 0 in R4R4

It is shown that a linear Hamiltonian system of signature zero on R4R4 is elliptic, hyperbolic or mixed according to the number of Lagrangian planes in the null-cone H−1(0)H−1(0), or equivalently the number of invariant Lagrangian planes. A weaker extension to higher dimensions is described.