En series of Lie algebras

The $E_n$ series of complex Lie algebras, in the convention relevant here, is the sequence

\begin{array}{c|c|c|c} n & E_n & \text{Dynkin type} & (\operatorname{rank},\dim)\\ \hline 3 & \mathfrak{sl}_3\oplus\mathfrak{sl}_2 & A_2\!+\!A_1 &(3,11)\\ 4 & \mathfrak{sl}_5 & A_4 &(4,24)\\ 5 & \mathfrak{so}_{10} & D_5 &(5,45)\\ 6 & \mathfrak e_6 & E_6 &(6,78)\\ 7 & \mathfrak e_7 & E_7 &(7,133)\\ 8 & \mathfrak e_8 & E_8 &(8,248). \end{array}

For $n=6,7,8$ this is the usual exceptional $E$ -family. The notation for $n=3,4,5$ is an extension convention, not a claim that those algebras are exceptional.

Regular inclusions

A compatible choice of roots gives a chain of semisimple regular subalgebras

\mathfrak{sl}_3\oplus\mathfrak{sl}_2 \subset\mathfrak{sl}_5 \subset\mathfrak{so}_{10} \subset\mathfrak e_6 \subset\mathfrak e_7 \subset\mathfrak e_8.

Combinatorially, successive terms are obtained by adding a node to a compatible Dynkin diagram, or in the reverse direction by deleting an appropriate node. The inclusions are additional structure: an abstract isomorphism class named in the table does not by itself select a particular embedded copy in the next algebra.

Useful adjacent branching patterns include

\begin{aligned} \mathfrak e_6&\downarrow\mathfrak{so}_{10}\oplus\mathbb C, &\mathbf{78}&=\mathbf{45}_0\oplus\mathbf1_0 \oplus\mathbf{16}_{3}\oplus\mathbf{16}^*_{-3},\\ \mathfrak e_8&\downarrow\mathfrak e_7\oplus\mathfrak{sl}_2, &\mathbf{248}&=(\mathbf{133},\mathbf1)\oplus(\mathbf1,\mathbf3) \oplus(\mathbf{56},\mathbf2). \end{aligned}

Charge signs and the labeling of dual spin modules depend on conventions.

Lower indices and convention warning

Extensions below $E_3$ are not uniform across the literature. A common convention sets $E_2$ to $A_1$ plus a one-dimensional abelian algebra and $E_1=A_1$ , but other contexts use different global groups, real forms, or extra $U(1)$ factors. For this reason, a bare symbol $E_n$ with $n<6$ should be accompanied by an explicit algebra or group.

The table concerns complex Lie algebras. Compact group versions require choices of global form and sometimes finite central quotients; one cannot recover those choices from a Dynkin diagram alone.

Paper context

The three-generation construction uses the truncated chain

\mathfrak{sl}_3\oplus\mathfrak{sl}_2 \subset\mathfrak{sl}_5\subset\mathfrak{so}_{10} \subset\mathfrak e_6\subset\mathfrak e_7

as a systematic root-removal framework. Its Standard Model algebra has an additional one-dimensional central summand, so $\mathfrak g_{\mathrm{SM}}=\mathbb C\oplus\mathfrak{sl}_2\oplus\mathfrak{sl}_3$ is reductive rather than equal to the semisimple $E_3$ term.

References

Nicolas Bourbaki, Lie Groups and Lie Algebras, Chapters 4--6, Springer, 2002, Plates I and IV--VII. Publisher record.
John C. Baez and John Huerta, Division Algebras and Supersymmetry II, Advances in Theoretical and Mathematical Physics 15 (2011), 1373--1410, Section 3. DOI.
John C. Baez, Three Generations in E7, 2026. arXiv:2608.06271.