The images of \(i,j,k\) satisfy the quaternion multiplication relations, so \(M\) is an injective real-algebra homomorphism. Direct calculation gives \[M(q)^*M(q)=|q|^2I,\qquad \det M(q)=|q|^2.\] Conversely, a special unitary matrix of degree two has the form \[\begin {pmatrix}z&w\\-\overline w&\overline z\end {pmatrix}, \qquad |z|^2+|w|^2=1,\] by orthonormality of its columns and determinant one. Taking real and imaginary parts of \(z,w\) gives a unique unit quaternion. This proves the group identification.

Conjugation preserves imaginary quaternions and their norm, so gives an orthogonal map of \(\mathbb R^3\). Its determinant is one: it depends continuously on the unit quaternion, the sphere \(S^3\) is connected, and at \(q=1\) it equals one. A quaternion commuting with all of \(i,j,k\) is real, so the kernel on unit quaternions is \(\{1,-1\}\). More explicitly, for a unit imaginary quaternion \(n\) and \(q=\cos (\theta /2)+n\sin (\theta /2)\), quaternion multiplication gives \[qvq^{-1}=v\cos \theta +(n\times v)\sin \theta +n(n\cdot v)(1-\cos \theta ).\] This is rotation through \(\theta \) about \(n\). Every element of \(SO(3)\) has an axis: its real orthogonal eigenvalues include one, and its restriction to the perpendicular plane is a planar rotation. The displayed formula therefore proves surjectivity.

The proper rotations of a regular solid act transitively on its vertices. A rotation fixing a vertex must rotate its incident edges cyclically about the center-to-vertex axis; all those cyclic rotations are symmetries. The tetrahedron has 4 vertices with stabilizer order three; the octahedron has 6 with stabilizer order 4; the icosahedron has 12 with stabilizer order 5. Orbit–stabilizer therefore gives \(|R_T|=12\), \(|R_O|=24\), \(|R_I|=60\). Every rotation has exactly two lifts under \(\pi \), giving the stated binary group orders and quotients.