The way I understand it, the dimension of a Lie group is the (minimal) number of real parameters required to label a single element of the group. I know that $SU(2)$ is a 3-dimensional group, but the argumentation I've read in some textbooks/references is sloppy. Here's how it usually goes.
Consider $h\in SU(2)$ a $2\times2$ complex matrix. Since $h^\dagger=h^{-1}$, we can parametrize $h$ this way: $$h=\begin{pmatrix} a & b \\ -b^* & a^* \end{pmatrix}.$$
Let $a = x+iy$ and $b=z+iw$ with $x,y,z,w\in \mathbb{R}$. The condition $\det h = 1$ implies that: $$x^2+y^2+z^2+w^2 =1.$$ Most references conclude here by saying that there are 4 real parameters that must satisfy the previous constraint, so the dimension is $4-1=3$.
My problem is that, using this parametrization, setting $x,y,z$ for instance is usually not sufficient to indicate a single matrix of $SU(2)$. Since the condition above is quadratic, then $w$ could take 2 values: $w=\pm \sqrt{1-x^2-y^2-z^2}$.
I know there are other parametrizations that don't include a 4th dependent variable, such as $h= \exp{i \alpha^i \sigma_i}$ (where $\sigma_i$ are the Pauli matrices) or $$ h = \begin{pmatrix} e^{i\alpha}\cos\theta & -e^{-i\beta}\sin\theta \\ e^{i\beta}\sin\theta & e^{-i\alpha}\cos\theta \end{pmatrix}.$$
Using these, it is much clearer that $SU(2)$ is 3-dimensional. But is it always necessary to find such a parametrization (without any dependent variable) to know the dimension of a Lie group? As I said, the proofs I read didn't care to go this far. They seemed to assume that any condition, even in a quadratic form as above, is sufficient to decrease the dimension of the group. Is this true or the authors of the proofs were just being sloppy?
My hypothesis is that perhaps they should have mentioned that the above condition is special because we recognize it as the equation of a 3-sphere (on which any point can be labeled using 3 angles), but that in general the dimension is not so trivial to find.