Abstract In this paper we study a slice-based notion of regularity for the split quaternionic algebra S S, whose natural quadratic form has signature (2, 2). For each hyperbolic unit J ∈ T J T (with J 2 = 1) we consider the slice L J = x + J y: x, y ∈ R L₉=\x+Jy: x, y R\ and define left s -regularity by the vanishing of the hyperbolic Cauchy–Riemann operator D J * D₉^{} on L J. We establish structural results that clarify how s -regularity behaves under orthogonal choices of units and how it interacts with power series on slices. In particular, we prove a splitting principle (Theorem 9) and a rigorous Abel-type boundary limit theorem for slice power series (Theorem 6), where non-tangential (Stolz-type) approach regions and an idempotent decomposition replace norm-based arguments that fail in the indefinite setting. We also provide a precise relation between the original M -approach condition and Stolz regions (Proposition 6. 1) and discuss the wave/ultra-hyperbolic character of the induced second-order operators. A comparison with quaternionic slice-regularity and a brief Schur-analysis outlook position the present theory within the broader landscape of hypercomplex analysis.
Ji Eun Kim (Thu,) studied this question.