A $S^1$ invariant frame on $S^3$

Question

We consider the action $S^1$ on $S^3$ with $\alpha.(z_1,z_2)=(\alpha z_1, \alpha z_2)$ where $\alpha \in S^1,\;(z_1,z_2)\in \mathbb{C}^2 \; \text{with} \; |z_1|^2+|z_2|^2=1$.

Are there three independent vector fields $S_1, S_2, S_3$ on $S^3$ such that each $S_i$ is invariant under action of $S^1$?

Answer 1

Use $S^3=Sp(1)\subset\mathbb{H}=\mathbb{C}\oplus\mathbb{C}j$ and right-multiply by $i,j,k$. Then the left multiplication by $S^1\subset\mathbb{C}$ commutes with the right-multiplication, so you have $S^1$-invariant.