G2 (split, connected adjoint)

The catalogue ${G_2}^{\mathrm{split}}_{\mathrm{ad},0}$ is the connected adjoint real Lie group

{G_2}^{\mathrm{split}}_{\mathrm{ad},0}=\operatorname{Aut}_{\mathbb R\text{-Lie}}(\mathfrak g_2^{\mathrm{split}})^0,

where $\mathfrak g_2^{\mathrm{split}}$ is the split real simple Lie algebra of Dynkin type $G_2$ and the superscript 0 denotes the identity component. This definition fixes the global form, rather than only its local Lie algebra.

Dimensions and structure

This group has real dimension $14$ . Its real rank is 2. It is centerless and noncompact. The familiar Killing-signature notation is $G_{2(2)}$ , but that notation alone does not specify an adjoint group, a finite central cover, or a topological universal cover. The catalogue uses the connected adjoint form throughout the split exceptional lane.

References

Pavel Etingof, Lie Groups and Lie Algebras (MIT lecture notes), §40.1, p. 213; §42.2, pp. 225–229; §43.1, Theorem 43.4, pp. 230–231.