Is there an explicit example of a left-invariant (unit)vector field associated with the Lie group $S^1$ (1-sphere)? That is, I am looking for a vector field $X:S^1\to\mathbb{R}^2$.
Moreover, is there an example of a left-invariant (unit)vector-field $Y:S^3\to\mathbb{R}^4$ associated with the Lie group $S^3$?
Remark: From the theory of Lie-groups it is known that such vector-field do exist on $S^1$ and $S^3$.