In this talk, we present a characterization of the Christoeffel pairs of timelike isothermic surfaces in the four-dimensional split-quaternions. When restricting the receiving space to the three-dimensional imaginary split-quaternions, we establish an equivalent condition for a timelike surface in $R^3_2$ to be real or complex isothermic in terms of the existence of integrating factors. This is joint work with M. Magid (Wellesley College).