The lie algebra su(2) is the linear combination of the pauli matrices. For instance : $x \sigma_x+y\sigma_y+z\sigma_z$. There are three degrees of freedom, namely x,y,z.
Let us compare it to the SU(2) group.
The definition of the SU(2) group is :
$$ SU(2):=\left\{ \pmatrix{\alpha & - \overline{\beta} \\ \beta &\overline{\alpha} } : \alpha,\beta \in \mathbb{C}, |\alpha|^2+|\beta|^2=1 \right\} $$
where $\alpha=a+ib$ and $\beta = c+id$. So the SU(2) has 4 degrees of freedom.
The algebra is connected to the group by the map $\exp^{ i su(2)} = SU(2)$
How can a lie algebra with 3 independent variables generate a group with 4 independent variables?
The condition $|\alpha|^2 +|\beta|^2=1$ reduce the degree of freedom by 1.