Intersections of complex structures

We study the sets of planes in an even dimensional real vector space $V$ which are simultaneously stabilised by a pair of complex structures on $V$. We completely describe these sets of planes for pairs of orthogonal complex structures. Generically, the number of such planes is finite. We compute this number for orthogonal complex structures and prove that it gives a lower bound for the number of planes simultaneously stabilised by a generic pair of complex structures on $V$.