  We give a simple interpretation of the adapted complex structure of
Lempert-Szoke and Guillemin-Stenzel: it is given by a polar decomposition of
the complexified manifold. We then give a twistorial construction of an
SO(3)-invariant hypercomplex structure on a neighbourhood of $X$ in $TTX$,
where $X$ is a real-analytic manifold equipped with a linear connection. We
show that the Nahm equations arise naturally in this context: for a connection
with zero curvature and arbitrary torsion, the real sections of the twistor
space can be obtained by solving Nahm's equations in the Lie algebra of certain
vector fields. Finally, we show that, if we start with a metric connection,
then our construction yields an SO(3)-invariant hyperk\"ahler metric.
