It is a basic and "intuition request" question. I have asked it on StackExchange yet it is probably to specialized for it since there were no answears.

Generalised complex structure is defined to be a field of endomorphisms $\mathcal{J}$ of the big tangent bundle $T^{big}M=TM \oplus T^*M$ such that $\mathcal{J}^2=-I$ and being orthogonal with respect to "natural paring" - neutral metric - $<X+ \xi, Y+\eta>=\xi(Y)+\eta(X)$. The first condition gives reduction of structure group of $T^{big}M$ to complex group, my question is why do we require the second condition?