The motivation of this question is here, where I asked about some kind of general parametrization for tori in $\mathbb{S}^3$.
Apparently not even the parametrization that I included in that question for tori in $\mathbb{R}^3$ is valid, as it was suggested in one answer. Therefore it looks very unlikely to find what I was asking for. Thus, I wonder if at least it is possible to construct tori of revolution in $\mathbb{S}^3$ by considering some smooth curve in $\mathbb{R}^4$ like \begin{align} C:I&\subset\mathbb{R}\longrightarrow\mathbb{R}^4\\ &t\longmapsto\begin{pmatrix} 0\\ 0\\ f(t)\\ g(t) \end{pmatrix}, \end{align} for some $f,g\in\mathcal{C}^\infty(\mathbb{R})$ with the condition $$f^2(t)+g^2(t)=1$$ to ensure that the curve is in $\mathbb{S}^3$ and rotate this curve in $\mathbb{R}^4$ by multiplying it by some rotational matrix of $\mathrm{SO}(4)$.
Questions:
Is this method valid to produce tori of revolution in $\mathbb{S}^3$ in a general way? (By general I just mean depending on the choice of functions $f$ and $g$ that define the generatrix curve). Or maybe should I also restrict this choice to closed curves?