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The one-parameter subgroup ${\gamma }_{\xi }(t)=\exp (t\xi )$ is the integral curve through the identity of $l_{\xi }$. Its defining initial-value problem is

For $\xi$ in the Lie algebra $\mathfrak{g}$, set ${\Phi }^t=R_{\exp (t\xi )}$. The one-parameter subgroup law gives the flow law, andso this is the global flow of the left-invariant vector field $l_{\xi }$.

For $f(g)=R_g^{\ast }\alpha$, every tangent vector at $g$ is the initial velocity of $g\exp (t\xi )$ for some $\xi$. Since $R_{g\exp (t\xi )}=R_{\exp (t\xi )}\circ R_g$,Thus all ${\mathcal{L}}_{l_{\xi }}\alpha$ vanish exactly when every derivative of $f$ vanishes. This is equivalent to $f$ being a locally constant function.

The matrices are rotations about the third coordinate axis. Direct multiplication givesand $g(\theta {)}^{\mathsf{T}}g(\theta )=I$. Thus they form a one-parameter subgroup of the orthogonal group $O(3)$, isomorphic to the circle group.