Why is $SU(2)$ a submanifold of $\mathbb{R}^8$?

Question

Let $SU(2) := \{A \in M_2(\mathbb{C}): \det A = 1, AA^\dagger = I_2 = A^\dagger A\}$.

Is this an embedded submanifold of $\mathbb{R}^8$? I tried the regular value theorem but this seems to be fruitless here. Should I use the definition with charts?

Answer 1

Hint: Use the defining equations to show that every $M \in \operatorname{SU}(2)$ is of the form $\begin{pmatrix} a & -\bar{b} \\ b & \bar{a} \end{pmatrix}$ for some $a, b \in \mathbb{C}$ with $|a|^2 + |b|^2 = 1$.

This normalization also allows you to set up a nice diffeomorphism between $S^3 \subset \mathbb{R}^4$ and $\operatorname{SU}(2)$.