Editing 3D rotation group (section)

=== Isomorphism with 𝖘𝖚(2) ===
The Lie algebras <math>\mathfrak{so}(3)</math> and <math>\mathfrak{su}(2)</math> are isomorphic. One basis for <math>\mathfrak{su}(2)</math> is given by<ref>{{harvnb|Hall|2015}} Example 3.27</ref>

:<math>\boldsymbol{t}_1 = \frac{1}{2}\begin{bmatrix}0 & -i\\ -i & 0\end{bmatrix}, \quad \boldsymbol{t}_2 = \frac{1}{2} \begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}, \quad \boldsymbol{t}_3 = \frac{1}{2}\begin{bmatrix}-i & 0\\ 0 & i\end{bmatrix}.</math>

These are related to the [[Pauli matrix|Pauli matrices]] by

:<math>\boldsymbol{t}_i \longleftrightarrow \frac{1}{2i} \sigma_i.</math>

The Pauli matrices abide by the physicists' convention for Lie algebras. In that convention, Lie algebra elements are multiplied by {{mvar|i}}, the exponential map (below) is defined with an extra factor of {{mvar|i}} in the exponent and the [[structure constant]]s remain the same, but the ''definition'' of them acquires a factor of {{mvar|i}}. Likewise, commutation relations acquire a factor of {{mvar|i}}. The commutation relations for the <math>\boldsymbol{t}_i</math> are

:<math>[\boldsymbol{t}_i, \boldsymbol{t}_j] = \varepsilon_{ijk}\boldsymbol{t}_k,</math>

where {{math|[[Levi-Civita symbol|''ε''<sub>''ijk''</sub>]]}} is the totally anti-symmetric symbol with {{math|1=''ε''<sub>123</sub> = 1}}. The isomorphism between <math>\mathfrak{so}(3)</math> and <math>\mathfrak{su}(2)</math> can be set up in several ways. For later convenience, <math>\mathfrak{so}(3)</math> and <math>\mathfrak{su}(2)</math> are identified by mapping

:<math>\boldsymbol{L}_x \longleftrightarrow \boldsymbol{t}_1, \quad \boldsymbol{L}_y \longleftrightarrow \boldsymbol{t}_2, \quad \boldsymbol{L}_z \longleftrightarrow \boldsymbol{t}_3,</math>

and extending by linearity.