Quaternionic left-invariant frame on the three-sphere
= Quaternionic left-invariant frame on the three-sphere
{title2=$q\cdot X_i=0,\quad X_i\cdot X_j=\delta_{ij}$}
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>.