SO(4,C) - Knowlpedia

$\operatorname{SO}(4,\mathbb C)$ is the special orthogonal group of $\mathbb C^{4}$ for the bilinear form $\sum_jx_jy_j$ :

\operatorname{SO}(4,\mathbb C)=\{A\in M_{4}(\mathbb C):A^{\mathsf T}A=I,\ \det A=1\}.

Over the complex numbers transpose does not mean conjugate transpose.

Dimensions and structure

This group has real dimension $12$ , complex dimension $6$ . Its tangent algebra is the skew-symmetric $4\times4$ matrices. The spin covering has kernel {±1}, so this group is not simply connected.

Its complex Lie group structure uses holomorphic multiplication and inversion. Forgetting the complex structure gives the underlying real Lie group; real and holomorphic homomorphisms belong to different categories.

References

Pavel Etingof, Lie Groups and Lie Algebras (MIT lecture notes), §31.2, Example 31.3 and Proposition 31.4; §31.3, Example 31.10, pp. 166–168.