In this talk, we address the problem of finding a sectionally Clifford algebra-valued harmonic function, vanishing at infinity, and satisfying certain boundary value conditions related to higher-order Lipschitz functions. Our main tools are the Hardy projections associated with a singular integral operator arising in bimonogenic function theory, which proves to be an involution operator on the first-order Lipschitz classes. This result generalizes the classical Hardy decomposition of Hölder continuous functions on a simple closed curve in the complex plane.