A general construction of alternative algebras with three anticommuting elements and a unit is given. As an exhaustive result over the real and complex fields, we obtain the Clifford algebrasH (quaternions), N 1 (dihedral Clifford algebra which is related to real 2‐spinors), and S 1(algebra of Pauli matrices which is related to complex 2‐spinors). What is important is that the algebrasN 1 and S 1 possess inverses everywhere except on a region akin to the light cone of the Minkowski space, while the quaternion algebraH has inverses everywhere except at the zero element. We discuss the reasons why the three algebrasN 1, H, and S 1 are so difficult to distinguish in the representation space of 2×2 complex matrices.