Why does SU(2) have 3 parameters/generators like the SO(3)?

[TimeRip496]

From "Symmetry and the Standard Model: Mathematics and Particle Physics by Matthew Robinson", it states that 'SU(2) matrix has one of the real parameters fixed,leaving three real parameters'.

I don't really get this part and hope someone can clear my doubt.
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To ensure that the matrix is unitary, c=-b* & d=a* while a=a and b=b. To ensure that it is special, ad-bc=1

Together, $$ ad-bc=a.a^*-b.(-b^*)=|a|^2 + |b|^2 = 1 $$

$$a=Re(a)+i Im(a)$$ & $$b=Re(b)+i Im(b)$$ ← From here I got 4 parameters.

$$ |a|^2 + |b|^2 = (Re(a))^2 +(Im(a))^2 + (Re(b))^2 +(Im(b))^2 = 1 $$← I still see 4 parameters thus I am still not sure how the unit determinant fix one of the parameter.

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Can I think of it another way as in just like SO(3) rotation, in SU(2) we also have a 3D space except instead of z axis, we have the i(imaginary) axis. So is no different from rotation in 3D space other than just changing the one of the axis to i-axis.

 

[fresh_42]

You can write $a=\pm \sqrt{1-b^2-c^2-d^2}$ which leaves you with three free variables $b,c,d$.

Or you can pass over to its Lie algebra $\mathfrak{su}(2)$, which is of the same dimension and has the $i-$multiples of the Pauli-matrices as real basis.

Or you can identify an element $\begin{bmatrix}x_1\mathbf{1}+x_2\mathbf{i} & -x_3\mathbf{1}+x_4\mathbf{i} \\x_3\mathbf{1}+x_4\mathbf{i} & x_1\mathbf{1}-x_2\mathbf{i} \end{bmatrix}$ of $SU(2)$ with a point $x_1\mathbf{1}+x_2\mathbf{i}+x_3\mathbf{j}+x_4\mathbf{k}$ in $\mathbb{R}^4$. Now the determinant condition reduces these point to the sphere $\mathbb{S}^3$, which is a three-dimensional manifold.