Complex Lie algebra so12(C)

The complex Lie algebra $\mathfrak{so}_{12}(\mathbb C)$ is the Lie algebra of infinitesimal transformations preserving a nondegenerate symmetric bilinear form on $\mathbb C^{12}$ . It is simple, of complex dimension $66$ , rank $6$ , and Dynkin type $D_6$ .

Its standard module has dimension $12$ . Its two half-spin modules $S^+$ and $S^-$ each have dimension $2^{6-1}=32$ , and its adjoint representation is $\Lambda^2\mathbb C^{12}$ , of dimension $66$ .

Groups and real forms

The half-spin modules integrate to the simply connected group $\operatorname{Spin}(12,\mathbb C)$ , but generally not to $SO(12,\mathbb C)$ . The compact real form integrates to compact $\operatorname{Spin}(12)$ ; the split real form is $\mathfrak{so}(6,6)$ . A notation such as $\mathfrak{so}_{12}$ in a complex-Lie-algebra calculation suppresses these global and real-form choices.

Role inside E7

There is a maximal-rank regular inclusion into the exceptional Lie algebra $\mathfrak e_7$ :

\mathfrak{sl}_2\oplus\mathfrak{so}_{12} \subset\mathfrak e_7.

Under this subalgebra, the adjoint representation branches as

\mathbf{133} \cong(\mathbf3,\mathbf1)\oplus(\mathbf1,\mathbf{66}) \oplus(\mathbf2,\mathbf{32}),

where $\mathbf{32}$ is one of the two half-spin modules; which one is called $S^+$ is conventional.

In the three-generation construction, each of three root $\mathfrak{sl}_2$ 's in a generation-symmetry $\mathfrak{sl}_3$ has a centralizer isomorphic to $\mathfrak{so}_{12}$ . Each associated $\mathfrak{sl}_2\oplus\mathfrak{so}_{12}$ contains a $32$ -dimensional subspace that restricts to one generation of Standard Model fermions and antifermions.

References

Nicolas Bourbaki, Lie Groups and Lie Algebras, Chapters 4--6, Springer, 2002, Plate IV. Publisher record.
William Fulton and Joe Harris, Representation Theory: A First Course, Springer, 1991, Sections 19--20. Publisher record.
John C. Baez, Three Generations in E7, 2026. arXiv:2608.06271.