  The subject of this paper is six-dimensional nearly (para-)K\"ahler geometry
with pseudo-Riemannian metrics. Firstly, we derive the analogue of the
well-known exterior differential system characterising a nearly K\"ahler
manifold and prove applications to the automorphism group of a nearly
(para-)K\"ahler structure. Secondly, we prove existence and uniqueness results
for left-invariant nearly (para-)K\"ahler structures on Lie groups $G \times G$
where $G$ is three-dimensional and simple.
