Quaternionic left-invariant frame on the three-sphere ($q\cdot X_i=0,X_i\cdot X_j={\delta }_{ij}$)

For $q=(q_0,q_1,q_2,q_3)\in S^3$, the vectors $(-q_1,q_0,q_3,-q_2)$, $(-q_2,-q_3,q_0,q_1)$ and $(-q_3,q_2,-q_1,q_0)$ are tangent and orthonormal. Under SU(2) as the three-sphere they are the left-invariant vector fields obtained by translating $-i$ times the Pauli matrices. This explicitly realizes the parallelization of a Lie group by left translations.