  Let T be the nilpotent group of 4 x 4 real upper triangular matrices. In this
note we show that the Euler equations of certain left-invariant riemannian
metrics on T have a horseshoe. We also show, with the aid of a numerical
computation of a Melnikov-type integral, that the Euler equations of the
sub-riemannian Carnot metric on T has a horseshoe. This sharpens an earlier
result of Montgomery, Shapiro and Stolin who had shown that the equations are
algebraically non-integrable.
