I lack the intuition to understand the answer of the following problem. Let the so called hopf fibration given by $h(x,y,z,t)=(x^2+y^2-z^2-t^2),2(yz-xt),2(xz+yt)$ and consider a vector field $X$ on $S^3$, namely $$X:=\frac{1}{2}\left(-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y}-t\frac{\partial}{\partial z}+z\frac{\partial}{\partial t}\right)$$ And now compute $\mathcal{L}_X(h_1),\mathcal{L}_X(h_2),\mathcal{L}_X(h_3)$.
What is bothering me, once I compute these, they all become $0$. My question is, why do they become zero and what does this mean?
