linguistic:voice_change, structural:expansion, linguistic:translation
Three-dimensional and four-dimensional rotation groups

The word " conjugation ", besides the meaning given above, can also mean taking an element a to r a r where r is some nonzero quaternion. All elements that are conjugate to a given element (in this sense of the word conjugate) have the same real part and the same norm of the vector part. (Thus the conjugate in the other sense is one of the conjugates in this sense.)

Thus the multiplicative group of nonzero quaternions acts by conjugation on the copy of R 3 {\displaystyle \mathbb {R} ^{3}} consisting of quaternions with real part equal to zero. Conjugation by a unit quaternion (a quaternion of absolute value 1) with real part cos(φ) is a rotation by an angle 2 φ, the axis of the rotation being the direction of the vector part. The advantages of quaternions are:

The set of all unit quaternions (versors) forms a 3-sphere S and a group (a Lie group) under multiplication, double covering the group SO (3, R) {\displaystyle {\text{SO}}(3,\mathbb {R})} of real orthogonal 3×3 matrices of determinant 1 since two unit quaternions correspond to every rotation under the above correspondence. See plate trick.

The image of a subgroup of versors is a point group, and conversely, the preimage of a point group is a subgroup of versors. The preimage of a finite point group is called by the same name, with the prefix binary. For instance, the preimage of the icosahedral group is the binary icosahedral group.

The versors' group is isomorphic to SU(2), the group of complex unitary 2×2 matrices of determinant 1.

Let A be the set of quaternions of the form a + b i + c j + d k where a, b, c, and d are either all integers or all half-integers. The set A is a ring (in fact a domain) and a lattice and is called the ring of Hurwitz quaternions. There are 24 unit quaternions in this ring, and they are the vertices of a regular 24 cell with Schläfli symbol {3,4,3}. They correspond to the double cover of the rotational symmetry group of the regular tetrahedron. Similarly, the vertices of a regular 600 cell with Schläfli symbol {3,3,5 } can be taken as the unit icosians, corresponding to the double cover of the rotational symmetry group of the regular icosahedron. The double cover of the rotational symmetry group of the regular octahedron corresponds to the quaternions that represent the vertices of the disphenoidal 288-cell.