g2 — split exceptional Lie algebra

The split real Lie algebra of type G2 is the real span of a Chevalley basis of $\mathfrak{g}_{2}(\mathbb C)$ , with the inherited bracket.

Construction and dimensions

The real dimension is 14 and the real split rank is 2. The basis has 2 Cartan directions and 12 root directions. Tensoring with $\mathbb C$ recovers the complex exceptional algebra. Different compatible Chevalley-basis choices give isomorphic real Lie algebras.

The common signature notation is $\mathfrak{g}_{2(2)}$ ; the parenthesized number is the number of positive minus negative Killing-form directions. It is not the dimension.

References

Knapp, Lie Groups Beyond an Introduction, digital second edition, §VI.1, Corollary 6.10 and Theorem 6.11, pp. 352–354; Appendix C.2, pp. 686–692.