The special unitary group

is a compact, connected, simply connected Lie group (topologically $S^3$).

Its Lie algebra is $\mathfrak{su}(2)$ , the traceless skew-Hermitian $2\times 2$ matrices.

A concrete basis and commutators

Using the Pauli matrices $\sigma_1,\sigma_2,\sigma_3$, set

A standard multiplication calculation yields

so (up to an overall sign convention) $\mathfrak{su}(2)$ has the same structure constants as $\mathfrak{so}(3)$ (compare the $\mathfrak{so}(3)$ example ).

Exponentials (explicit)

For $t\in\mathbb R$, [ \exp(tX_3)=\exp!\left(\frac{it}{2}\begin{pmatrix}1&0\0&-1\end{pmatrix}\right)

\begin{pmatrix}e^{it/2}&0\0&e^{-it/2}\end{pmatrix}\in SU(2). ] This shows directly that one-parameter subgroups arise from the exponential map .

The 2-to-1 covering $SU(2)\to SO(3)$

Let $SU(2)$ act on $\mathfrak{su}(2)$ by conjugation $g\cdot X=gXg^{-1}$. This preserves the (negative) Killing form inner product, giving a homomorphism

(compare adjoint action ). One checks:

• $\mathrm{Ad}$ is surjective,
• $\ker(\mathrm{Ad})=\{\pm I\}$.

Hence $\mathrm{Ad}$ is a covering Lie group map $SU(2)\to SO(3)$ with discrete central kernel $\{\pm I\}$.