SO(8,C) - Knowlpedia

$\operatorname{SO}(8,\mathbb C)$ is the special orthogonal group of $\mathbb C^{8}$ for the bilinear form $\sum_jx_jy_j$ :

\operatorname{SO}(8,\mathbb C)=\{A\in M_{8}(\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 $56$ , complex dimension $28$ . Its tangent algebra is the skew-symmetric $8\times8$ matrices. The spin covering has kernel {±1}, so this group is not simply connected. Dimension eight is the setting for triality; the vector and two half-spin representations live naturally on the spin group.

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.