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In this paper we revisit the concept of conformality in the sense of Gauss inthe context of octonions and Clifford algebras. We extend a characterization ofconformality in terms of a system of partial differential equations anddifferential forms using special orthonormal sets of continuous functions thathave been used before in the particular quaternionic setting. The aim is todescribe to which higher dimensional algebras this characterization can exactlybe extended and under which circumstances. It turns out to be crucial that thischaracterization requires a domain of definition that lies in a subalgebra thathas the norm composition property and that is either associative (Cliffordalgebra case) or at least alternative (octonionic case). The orthonormal framesare elements of the spin group Spin(n+1). We round off by relating the natureof the orthonormal frames to the associated M\"obius transformation which arerelated to SO(9,1) in the octonionic case and to the Ahlfors-Vahlen group inthe case of a Clifford algebra.

Conformal mappings revisited in the octonions and Clifford algebras of arbitrary dimension