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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.

For a Matrix Lie group, the right-invariant vector field is $r_{\xi }(g)=\xi g$. Differentiating these ambient matrix-valued functions in the definition of the Lie bracket of vector fields gives $[r_{\xi },r_{\eta }](g)=(\eta \xi -\xi \eta )g$. Thus $\xi \mapsto r_{\xi }$ is an antihomomorphism, and $\xi \mapsto -r_{\xi }$ is a homomorphism. In contrast, $\xi \mapsto l_{\xi }$ for left-invariant vector fields is a homomorphism.

The multiplication $(x^{\prime },y^{\prime },z^{\prime })(x,y,z)=(x^{\prime }+e^{-z^{\prime }}x,y^{\prime }+e^{z^{\prime }}y,z^{\prime }+z)$ defines the three-dimensional Sol Lie group. Its left-invariant vector fields $e^{-z}{\partial }_x,e^z{\partial }_y,{\partial }_z$ have dual coframe $e^{z}dx,e^{-z}dy,dz$. Their sum of squares is a left-invariant metric, $d{s}^2=e^{2z}d{x}^2+e^{-2z}d{y}^2+d{z}^2$.

Declare a global frame of left-invariant vector fields parallel on a semisimple Lie group. This affine connection is flat and preserves a left-invariant metric tensor because its frame components are constant. If $[e_a,e_c]=C_a{}^b{}_c{e}_b$, its torsion tensor is $T_a{}^b{}_c=-C_a{}^b{}_c$. For fixed $c$, the matrix with entries $T_a{}^b{}_c$ is $\mathrm{ad}{e}_c$. Its trace vanishes because a semisimple Lie algebra is perfect and adjoints of brackets are commutators. The trace of the product for $c,d$ is the Killing form $K_{cd}$, proving both contractions. This flat metric connection differs from the curved torsion-free Levi-Civita connection of the Killing metric.