  It is shown that, in four dimensions, it is possible to introduce coordinates
so that an analytic metric locally takes block diagonal form. i.e. one can find
coordinates such that $g_{\alpha\beta} = 0$ for $(\alpha, \beta) \in S$ where
$S = {(1, 3), (1, 4), (2, 3), (2, 4)}$. We call a coordinate system in which
the metric takes this form a 'doubly biorthogonal coordinate system'. We show
that all such coordinate systems are determined by a pair of coupled
second-order partial differential equations.
