Timelike Christoffel pairs in the split-quaternions

We characterize the Christoffel 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(2)(3) to be real or complex isothermic in terms of the existence of integrating factors.Institute of Mathematics and Statistics (IME) at the University of Sao Paulo (USP)Institute of Mathematics and Statistics (IME) at the University of Sao Paulo (USP)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)State of Sao Paulo Research Foundation (FAPESP