These “dihedrons” (which you might as well just call “2x2 matrices with real entries”) are isomorphic to the geometric algebra of the 2-dimensional Euclidean vector plane with signature (+, +), with 1 scalar component, 2 vector components, and 1 bivector component. Personally I find the basis 1, e₁, e₂, e₁e₂ and the notation of GA to be conceptually much clearer than the matrix entries and standard matrix sum/product to work with, but sometimes using a matrix representation is convenient in a computer.

(2x2 real matrices are also isomorphic to the geometric algebra of the 2-dimensional pseudo-Euclidean vector plane with signature (+, -), under a different interpretation.)

By comparison the quaternions are the even subalgebra of the geometric algebra of 3-dimensional Euclidean vector space with signature (+,+,+), consisting of only the 1 scalar and 3 bivector components. Or under a different interpretation are isomorphic to the full geometric algebra of the 2-dimensional vector plane with signature (-, -). They can be represented as Pauli matrices.

For more on this see the papers and books of Garret Sobczyk, https://garretstar.com/secciones/publications/publications.h... ; for example the recent https://www.garretstar.com/sobczyk09-mar-2020.pdf