Understanding vector field(s) on $\mathbb{S}^3$. I was slving the exercises of John Lee's book "Introduction to Smooth Manifolds", where there is an exercise asking us to prove that $\mathbb{S}^3$ is parallelizable. In the hint, the author asks us to consider the vector fields:
$$X_1 = -x\dfrac{\partial}{\partial w} + w \dfrac{\partial}{\partial x} - z \dfrac{\partial}{\partial y} + y \dfrac{\partial}{\partial z},$$
$$X_2 = -y\dfrac{\partial}{\partial w} + z \dfrac{\partial}{\partial x} + w \dfrac{\partial}{\partial y} - x \dfrac{\partial}{\partial z},$$
$$X_3 = -z\dfrac{\partial}{\partial w} - y \dfrac{\partial}{\partial x} + x \dfrac{\partial}{\partial y} + w \dfrac{\partial}{\partial z}.$$
I get the hint and how to use it. What I don't understand is why are the vector fields $4$-dimensional? Isn't $\mathbb{S}^3$ a $3$-dimensional manifold? This is why the tangent vectors should have only $3$ coordinates! I also searched other places on the internet and more or less, everybody uses $4$ coordinates for a vector field on $\mathbb{S}^3$. Could anybody help me understand this?

You can parametrize the sphere $S^3$ by polar coordinates but computations will be painful!
You can identify $S^3$ to the Lie group $SU(2)$ and find a basis to its Lie algebra. Lie groups are parallelizable (pick any basis of left invariant vector fields).
