  The main aim of this paper is to simplify and popularise the construction
from the 2013 paper by Apostolov, Calderbank, and Gauduchon, which (among other
things) derives the Plebanski-Demianski family of solutions of GR using ideas
of complex geometry. The starting point of this construction is the observation
that the Euclidean versions of these metrics should have two different
commuting complex structures, as well as two commuting Killing vector fields.
After some linear algebra, this leads to an ansatz for the metrics, which is
half-way to their complete determination. Kerr metric is a special 2-parameter
subfamily in this class, which makes these considerations directly relevant to
Kerr as well. This results in a derivation of the Kerr metric that is
self-contained and elementary.
