It is known that Clifford (geometric) algebra offers a geometric interpretation for square roots of –1 in the form of blades that square to –1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Research has been done [1] on the biquaternion roots of –1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra Cℓ 3 of $${{\mathbb R^3}}$$ . All these roots of –1 find immediate applications in the construction of new types of geometric Clifford Fourier transformations.
We now extend this research to general algebras Cℓ p,q . We fully derive the geometric roots of –1 for the Clifford (geometric) algebras with p + q ≤ 4.