  We investigate the erbium flip-flop dynamics as a limiting factor of the
electron spin lifetime and more generally as an indirect source of decoherence
in rare-earth doped insulators. Despite the random isotropic arrangement of
dopants in the host crystal, the dipolar interaction strongly depends on the
magnetic field orientation following the strong anisotropy of the $g$-factor.
In Er$^{3+}$:Y$_2$SiO$_5$, we observe by transient optical spectroscopy a three
orders of magnitude variation of the erbium flip-flop rate (10ppm dopant
concentration). The measurements in two different samples, with 10ppm and 50ppm
concentrations, are well-supported by our analytic modeling of the dipolar
coupling between identical spins with an anisotropic $g$-tensor. The model can
be applied to other rare-earth doped materials. We extrapolate the calculation
to Er$^{3+}$:CaWO$_4$, Er$^{3+}$:LiNbO$_3$ and Nd$^{3+}$:Y$_2$SiO$_5$ at
different concentrations.
