  Generalising work of Calaque-Enriquez-Etingof, we construct a universal KZB
connection D_R for any finite (reduced, crystallographic) root system R. D_R is
a flat connection on the regular locus of the elliptic configuration space
associated to R, with values in a graded Lie algebra t_R with a presentation
with relations in degrees 2, 3 and 4 which we determine explicitly. The
connection D_R also extends to a flat connection over the moduli space of
pointed elliptic curves. We prove that its monodromy induces an isomorphism
between the Malcev Lie algebra of the elliptic pure braid group P_R
corresponding to R and t_R, thus showing that P_R is not 1-formal and extending
a result of Bezrukavnikov valid in type A. We then study one concrete
incarnation of our KZB connection, which is obtained by mapping t_R to the
rational Cherednik algebra H_{h,c} of the corresponding Weyl group W. Its
monodromy gives rise to an isomorphism between appropriate completions of the
double affine Hecke algebra of W and H_{h,c}.
