Part 2 of 9. Every root carries a copy of sl2(ℂ), whose representation theory forces an integer constraint on every pair of roots. Reflections in the roots then generate a finite group, the Weyl group.
Part 1 put a Euclidean inner product on the real span of the roots but said nothing yet about how the roots can sit. The constraint comes from a copy of sl2(ℂ) attached to each root α, whose Cartan element can only have integer eigenvalues. This part shows that every root α carries a reflection sα: 𝔼 → 𝔼 that permutes Δ, and these reflections generate the Weyl group W. In E8, the eight simple reflections applied to one root reach all 240.
Fix α ∈ Δ. By the four facts of Part 1, −α ∈ Δ, both 𝔤α and 𝔤−α are one-dimensional, and [𝔤α, 𝔤−α] ⊆ 𝔥. Pick nonzero eα ∈ 𝔤α and fα ∈ 𝔤−α; their bracket [eα, fα] is some nonzero hα ∈ 𝔥, equal up to normalisation to the dual Cartan element tα.
Rescaling eα and fα makes (eα, fα, hα) satisfy the standard sl2(ℂ) relations:
So 𝔤α ⊕ 𝔤−α ⊕ ℂ·hα is a Lie subalgebra of 𝔤 isomorphic to sl2(ℂ), with H ↦ hα, E ↦ eα, F ↦ fα. There is one embedded sl2 per pair {α, −α}, and these |Δ|/2 subalgebras together cover every direction in 𝔤 outside the Cartan. Facts about sl2 therefore transport back to 𝔤, and the one needed here is integrality.
We use one fact about sl2(ℂ) without proof; it is a standard exercise with E raising and F lowering between H-eigenspaces. Every finite-dimensional complex representation V of sl2(ℂ) is a direct sum of H-eigenspaces with integer eigenvalues, and those eigenvalues are symmetric about 0: if k is an H-eigenvalue on V, so is −k, with the same eigenspace dimension.
The adjoint action makes 𝔤 a representation of sl2(α), with the basis acting as adhα, adeα, adfα. So the eigenvalues of adhα are integers.
Those eigenvalues are the values of the other roots β on hα: for y ∈ 𝔤β, [hα, y] = β(hα) · y, so β(hα) ∈ ℤ for every pair α, β.
Through 𝔥 ≅ 𝔥*, β(hα) is the Cartan integer of β against α,
where the factor of 2 and the denominator come from normalising hα so its eigenvalues are integers rather than half-integers. That nβ,α is an integer for every ordered pair is the first hard constraint on Δ.
The Cartan integers feed into a reflection. Fix α ∈ Δ and define sα: 𝔼 → 𝔼 by
This sα is the Euclidean reflection across the hyperplane orthogonal to α, since sα(α) = −α and sα(ξ) = ξ for ξ ⊥ α. It is linear, it is its own inverse, and sα(β) differs from β by the integer multiple −nβ,α of α. The same sl2 representation theory shows more: sα(β) is itself a root.
So each root gives a symmetry of Δ. This closure is a property of the root system; the formula alone would not guarantee it.
The Weyl group W is the subgroup of the orthogonal group O(𝔼) generated by all the reflections sα for α ∈ Δ:
W permutes Δ, and since Δ spans 𝔼 an element of W is determined by that permutation, so W embeds in the symmetric group on |Δ| letters and is finite. Every simple Lie algebra over ℂ therefore determines a finite set Δ ⊂ 𝔼 with integer Cartan integers, closed under the finite reflection group W. Classifying simple Lie algebras comes down to classifying such sets.
The orbit of α is the smallest W-stable set containing it, found by applying reflections until nothing new appears. In a simply-laced root system, where all roots have the same length, W acts transitively and there is one orbit. A2 (the sl3 hexagon) and all the ADE types are simply-laced. In a non-simply-laced system such as B2, C2, B3, G2 or F4, reflections preserve length, so there are two orbits, the long roots and the short roots.
Figure 1. Weyl orbit explorer. Click a root to highlight its orbit. In A2 every click lights the whole hexagon. B2 splits into the four long and the four short roots; G2 into six short and six long, with length ratio √3. Dashed grey lines are the hyperplanes of the generating reflections.
The Cartan integers of B2 can be computed by hand. The simple roots are α1 = e1 − e2 (long, squared length 2) and α2 = e2 (short, squared length 1):
The off-diagonal Cartan integers are −2 and −1, integers but unequal, which happens only when the two roots have different lengths. Their product nα₁, α₂ · nα₂, α₁ = 2 controls the angle via nα, β · nβ, α = 4 cos²θ.
For E8 all 240 roots have squared length 2, so sα(x) = x − ⟨x, α⟩ · α. For any simple root αi, the reflection fixes the 126 roots perpendicular to αi and moves the remaining 114.
Figure 2 draws each root as a polyline across eight parallel axes, one per coordinate.
Figure 2. One simple reflection splits the 240 roots of E8 into the fixed (grey) and the moved (amber). A root is fixed exactly when it is perpendicular to the chosen αi, and for every i that is 126 roots.
Start from α1 and apply simple reflections in any order. Because E8 is simply-laced, the orbit grows to all 240 roots.
Figure 3. The orbit generator in E8. Each click applies a simple reflection to every root collected so far and adds the new ones. Whatever the starting root and the order of generators, the orbit ends at 240. The word below records the sequence.
Every word in s1, …, s8 is a symmetry of E8, and the set of all of them is W(E8), of order 696,729,600 = 214 · 35 · 52 · 7. For comparison, the Weyl group of A8 is the symmetric group S9, of order 9! = 362,880, and that of D8 has order 27 · 8! = 5,160,960. W(E8) is 1,920 times larger than the first and 135 times larger than the second.
Every element is a product of at most 120 simple reflections, one for each positive root; 120 is the length of the longest element.
Integrality is not the only constraint. Since nα, β · nβ, α = 4 cos²θ ∈ [0, 4], the Cartan integers lie in {0, ±1, ±2, ±3, ±4}. Part 3 turns that into the crystallographic restriction: only four angles between roots are possible.