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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 ℝ³₂ to be real or complex isothermic in terms of the existence of integrating factors.
Lorentzian surfaces in Lorentz three-space are studied using an indefinite version of the quaternions. A classification theorem for Bonnet pairs in Lorentz three-space is obtained.
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