= Solution
The defining matrix conditions are
$$
\boxed{SU(2)=\{U\in M_2(\mathbb C):U^\dagger U=I,\ \det U=1\}}.
$$
The group operation is matrix multiplication. Equivalently $U=a_0I-i\mathbf a\cdot\boldsymbol\sigma$ with real $a_0,\mathbf a$ and $a_0^2+|\mathbf a|^2=1$, so it has three real parameters and is a compact connected simply connected <Lie group>. Its Hermitian <spin> generators obey the <commutator> from part (a); the corresponding Lie-algebra matrices themselves are traceless and anti-Hermitian.
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