Part 6 of 15. A, B, C, D. Before we prove the classification terminates, we instantiate the Dynkin machinery on the four infinite families we can build by hand: special linear, odd and even orthogonal, and symplectic Lie algebras. They give us the raw material that the termination proof in the next explainer will keep.
We have the full vocabulary now: Cartan subalgebras, root space decompositions, Killing forms, Weyl groups, simple roots, Cartan matrices, Dynkin diagrams. The classification reduces to one question: which connected Dynkin diagrams have positive-definite Cartan matrices? Before running the termination argument, we instantiate the Dynkin machinery on four matrix Lie algebra families.
The families An, Bn, Cn, Dn correspond to sln+1(ℂ), so(2n+1, ℂ), sp(2n, ℂ), and so(2n, ℂ). Each has a diagonal Cartan, a root system listed in coordinate functionals, and a distinctive Dynkin shape. The termination proof will show that only these plus E6, E7, E8, F4, G2 can occur.
sln+1(ℂ), the (n+1) × (n+1) trace-zero complex matrices, is the most familiar simple Lie algebra. Its Cartan is the n-dimensional space of diagonal trace-zero matrices. Its n(n+1) roots are
where εi(H) = hi. Simple roots αi = εi − εi+1 produce a Cartan matrix with −1 on the sub/super-diagonal and 0 elsewhere:
The Dynkin diagram is a linear chain of n single-edged nodes. An starts at n = 1: A1 = sl2.
so(2n+1, ℂ), the (2n+1) × (2n+1) skew-symmetric complex matrices, has dimension n(2n+1). The Cartan is an n-dimensional space of block-diagonal skew-symmetric matrices, the span of n commuting rotation generators. Its root system is
2n(n − 1) long roots (squared length 2) and 2n short roots (squared length 1). Simple roots αi = εi − εi+1 for i < n and αn = εn give a Cartan matrix matching An's except (n−1, n) = −2 and (n, n−1) = −1. The Dynkin diagram is a linear chain of n nodes with the last edge doubled, arrow outward.
Bn starts at n = 2, because so(3) ≅ sl2 means B1 = A1.
sp(2n, ℂ), the symplectic Lie algebra preserving a non-degenerate alternating form, has dimension n(2n+1) — same as so(2n+1, ℂ), but the two are non-isomorphic for n ≥ 3. The Cartan is a block-diagonal space, and the roots are
2n(n − 1) short roots (squared length 2) and 2n long roots (squared length 4). The simple roots are αi = εi − εi+1 and αn = 2εn. The Dynkin diagram is Bn's with the arrow reversed.
Cn starts at n = 3. C1 = A1, and C2 ≅ B2 (so(5) ≅ sp(4)).
so(2n, ℂ), the even-dimensional orthogonal Lie algebra, has dimension n(2n − 1). Its root system is
2n(n − 1) roots ±εi ± εj, all squared length 2. Dn is simply-laced. The simple roots are αi = εi − εi+1 and αn = εn−1 + εn. The Dynkin diagram is a chain with a fork at the end, branching at αn−2.
Dn starts at n = 4. D1 is abelian; D2 = A1 ⊕ A1; D3 = A3. D4 = so(8) is famous for triality: its diagram has three arms of length 1 at a central node, and the diagram symmetry group is S3.
Figure 1. The four classical families at variable rank. Each row shows a family's Dynkin diagram, the underlying matrix Lie algebra, and its structural invariants, algebra dimension (rank + |Δ|), root count, Coxeter number h, and Weyl group order |W|. At rank 2 and 3 some families start to coincide or become reducible, which is why the classification starts each family at its lowest irreducible rank: A from rank 1, B and C from rank 2 and 3 respectively, D from rank 4.
At low ranks the families coincide, which is why each starts at a specific rank:
| Diagram | Coincidence | Reason |
|---|---|---|
| B1 = A1 | so(3) ≅ sl(2) | 3-dim matching via cross-product iso (explainer 1) |
| C1 = A1 | sp(2) ≅ sl(2) | The 2×2 matrices preserving a form are also trace zero. |
| C2 ≅ B2 | sp(4) ≅ so(5) | Ten-dimensional double cover; same Dynkin, arrow flipped. |
| D1 | so(2), abelian | 1-dimensional rotations commute, not simple. |
| D2 = A1 ⊕ A1 | so(4) ≅ sl(2) × sl(2) | Splits into two commuting factors, not simple. |
| D3 = A3 | so(6) ≅ sl(4) | 15-dimensional identification via spinor construction. |
Figure 2. Low-rank coincidences among the classical families. The convention "Bn starts at 2" is really "B1 has already been listed under A1." These coincidences are the reason a naive enumeration by rank would overcount, and the reason the classification uses the ranges n ≥ 1 (A), n ≥ 2 (B), n ≥ 3 (C), n ≥ 4 (D) to avoid collisions.
All four classical families share a few structural features that matter for the classification:
The raw material. An, Bn, Cn, Dn are the four obvious Dynkin shapes from matrix constructions. The next explainers show that the only other shapes are the five exceptions G2, F4, E6, E7, E8.