  We investigate the $E_g \otimes e_g$ Jahn-Teller system for the purpose to
reveal the nature of quantum chaos in crystals. This system simulates the
interaction between the nuclear vibrational modes and the electronic motion in
non-Kramers doublets for multiplets of transition-metal ions. Inclusion of the
anharmonic potential due to the trigonal symmetry in crystals makes the system
nonintegrable and chaotic. Besides the quantal analysis of the transition from
Poisson to Wigner level statistics with increasing the strength of
anharmonicity, we study the effect of chaos on the electronic orbital angular
momentum and explore the magnetic $g$-factor as a function of the system's
energy. The regular oscillation of this factor changes to a rapidly-decaying
irregular oscillation by increasing the anharmonicity (chaoticity).
