Proving that $SU(2)$ is a differential submanifold of $\mathbb{C}^{4}$

Question

I would like to prove that the special unitary group $SU(2)$ is a $3$-dimensional compact submanifold of $\mathbb{C}^{4}$. How could I do it? I already proved that there exists a diffeomorphism between $S^3$ and $SU(2),$ so $SU(2)$ is compact. But that doesn't help me to continue.

Answer 1

Hint: $SU(2)\subset M(2,C)$, consider $sU(2)=\{M\in M(2,C), M+\bar M^t=0\}$ and $exp:su(2)\rightarrow M(2,C)$ the restriction of the exponential the image of $exp$ is $SU(2)$ and $exp$ is an immersion, and define a parameterization of $S(2)$ which shows that it is a 3-dimensional submanifold since the dimension of $su(2)$ is 3.