  The negative part $U^-$ of a quantised enveloping algebra associated to a
simple Lie algebra possesses a canonical basis $\mathcal{B}$ with favourable
properties. Lusztig has associated a cone to a reduced expression $\mathbf{i}$
for the longest element $w_0$ in the Weyl group of $\mathfrak{g}$, with good
properties with respect to monomial elements of $\mathcal{B}$. The first author
has associated a subalgebra $A_{\mathbf{i}}$ of $U^-$, compatible with the dual
basis $\mathcal{B}^*$, to each reduced expression $\mathbf{i}$. We show that,
after a certain twisting, the string parametrisation of the adapted basis of
this subalgebra coincides with the corresponding Lusztig cone. As an
application, we give explicit expressions for the generators of the Lusztig
cones.
