Linear Algebra Question in a Complex Geometry Proof

1. Jul 21, 2009

cogito²

My brain has ceased functioning while trying to understand a small aspect of the proof of Wirtinger's inequality on page 175 in Demailly's book: (http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf).

On the second line of that proof he says to "choose an oriented R-basis such that blah blah blah..." Basically he chooses an R-basis (e_1, e_2, ...) which is orthonormal wrt the real part of the hermitian metric, such that the imaginary part of the metric is non-zero except for in pairs. What I mean is that Im h(e_1, e_2), Im h(e_3,e_4), ... can be non-zero, but all other pairs (for example Im h(e_1,e_3)) are zero.

I'm having trouble seeing how this is done. I mean if I knew that the real submanifold were complex analytic ahead of time I could do it (which is done a little later in the proof), but I'm confused otherwise...this is basically just a linear algebra question so I just figured if someone here just sees this clearly they might save me a lot of head ache.

My explanation of the problem is pretty bad, but I figure since it's just the first couple lines in the proof, it might be easier to let the pdf to the talking....though if any of you guys disagree, I can try to clarify the question...