  We consider the free nilpotent Lie algebra $L$ with 2 generators, of step 4,
and the corresponding connected simply connected Lie group $G$. We study the
left-invariant sub-Riemannian structure on $G$ defined by the generators of $L$
as an orthonormal frame.
  We compute two vector field models of $L$ by polynomial vector fields in
$R^8$, and find an infinitesimal symmetry of the sub-Riemannian structure.
Further, we compute explicitly the product rule in $G$, the right-invariant
frame on $G$, linear on fibers Hamiltonians corresponding to the left-invariant
and right-invariant frames on $G$, Casimir functions and co-adjoint orbits on
$L^*$.
  Via Pontryagin maximum principle, we describe abnormal extremals and derive a
Hamiltonian system $\dot \lambda = \vec{H}(\lambda)$, $\lambda \in T^*G$, for
normal extremals.
  We compute 10 independent integrals of $\vec{H}$, of which only 7 are in
involution. After reduction by 4 Casimir functions, the vertical subsystem of
$\vec{H}$ on $L^*$ shows numerically a chaotic dynamics, which leads to a
conjecture on non-integrability of $\vec{H}$ in the Liouville sense.
