  The origin of the cooperative Jahn-Teller distortion and orbital-order in
LaMnO3 is central to the physics of the manganites. The question is complicated
by the simultaneous presence of tetragonal and GdFeO3-type distortions and the
strong Hund's rule coupling between e_g and t_2g electrons. To clarify the
situation we calculate the transition temperature for the Kugel-Khomskii
superexchange mechanism by using the local density approximation+dynamical
mean-field method, and disentangle the effects of super-exchange from those of
lattice distortions. We find that super-exchange alone would yield T_KK=650 K.
The tetragonal and GdFeO3-type distortions, however, reduce T_KK to 550 K. Thus
electron-phonon coupling is essential to explain the persistence of local
Jahn-Teller distortions to at least 1150 K and to reproduce the occupied
orbital deduced from neutron scattering.
