I am searching the polynomial functions from $S^3$ to $S^3$.

($S^3$ is the set of vectors $x$ in $\mathbb{R}^4$ such that $\|x\|=1$)

We say $g$ is a polynomial function from $S^3$ to $S^3$, if there exists $f_1,f_2,f_3,f_4 \in \mathbb{R}[X_1,X_2,X_3,X_4]$ and $f:\mathbb{R}^4 \rightarrow \mathbb{R}^4$, $(x_1,x_2,x_3,x_4) \mapsto (f_1(x_1,x_2,x_3,x_4),...,f_4(x_1,x_2,x_3,x_4))$ such that $g: S^3 \rightarrow S^3$, maps $(x_1,x_2,x_3,x_4)$ to $ f(x_1,x_2,x_3,x_4)$

We generate the set $F$ of functions from $S^3$ to $S^3$ by:

the constant functions belong to $F$,

the identity belongs to $F$,

the isometries belong to $F$,

if $f,g \in F$, $\overline{f} \in F$, and $f \circ g\in F$, and $ f \times g\in F$, where $\overline{z}$ is the conjuguate of $z$ in the quaternions, and $\times$ is the quaternions product.

Do all the polynomial functions from $S^3$ to $S^3$ belong to $F$ ?

We have identified $S^3$ with the set of quaternions $z$ such that $|z|=1$

Thanks in advance.