Exceptional Lie algebra g2

The exceptional complex Lie algebra $\mathfrak g_2$ is the unique simple complex Lie algebra whose root system has Dynkin type $G_2$ . It has complex dimension $14$ , rank $2$ , and $12$ roots, with squared-length ratio $3$ between long and short roots.

Its smallest nontrivial irreducible representation is the $7$ -dimensional fundamental module $\mathbf7$ . Its adjoint representation is $\mathbf{14}$ , and

\Lambda^2\mathbf7\cong\mathbf7\oplus\mathbf{14}.

Octonionic realization

For the complexified octonion algebra $\mathbb O_{\mathbb C}$ ,

\mathfrak g_2\cong\operatorname{Der}(\mathbb O_{\mathbb C}).

Derivations kill the identity and act irreducibly on the seven-dimensional imaginary, or trace-zero, subspace. This is the module $\mathbf7$ . Equivalently, $G_2$ is characterized inside $GL(7,\mathbb C)$ as the stabilizer of the generic alternating three-form induced by octonion multiplication.

Groups and real forms

The simply connected and adjoint complex groups of type $G_2$ coincide and have trivial center. The automorphism group of the real division octonions is the compact exceptional group $G_2$ . The split octonions instead yield the split real form $G_{2(2)}$ .

Paper context

Compact $G_2=\operatorname{Aut}(\mathbb O)$ acts on the choices of complex subalgebra $\mathbb C\subset\mathbb O$ . The stabilizer of a chosen imaginary unit is isomorphic to $SU(3)$ , and under it

\mathbb O\cong\mathbb C\oplus\mathbb C^3

as real vector spaces equipped with compatible complex structure. This supplies the $SU(3)$ -invariant inner product and conjugated cross product used in the exceptional-Jordan-algebra construction of octonion multiplication.

References

Nicolas Bourbaki, Lie Groups and Lie Algebras, Chapters 4--6, Springer, 2002, Plate IX. Publisher record.
John C. Baez, "The Octonions," Bulletin of the American Mathematical Society 39 (2002), 145--205. DOI.
John Frank Adams, Lectures on Exceptional Lie Groups, University of Chicago Press, 1996, Chapters 3--5. Publisher record.
John C. Baez and Paul Schwahn, The Standard Model Gauge Group from the Exceptional Jordan Algebra, 2026. arXiv:2606.15235.