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The last explainer put a Euclidean inner product on the real span of the roots, but we have no constraint yet on how those vectors can be arranged. The constraint comes from one observation: the root space decomposition packages every root α with an embedded copy of sl₂(ℂ) inside 𝔤, and sl₂ representation theory forces the eigenvalues of that sl₂'s Cartan element to be integers. From this integrality, the Weyl reflections emerge.

By the end of this explainer every root α will carry a reflection s_α: 𝔼 → 𝔼 that permutes Δ, and the group generated by all these reflections is the Weyl group W. In E₈, a single simple reflection iterated is enough to reach all 240 roots from one seed.

One sl₂ per root

Fix α ∈ Δ. From the four facts, −α ∈ Δ, both 𝔤_α and 𝔤_−α are one-dimensional, and [𝔤_α, 𝔤_−α] ⊆ 𝔥. Pick nonzero e_α ∈ 𝔤_α and f_α ∈ 𝔤_−α; their bracket [e_α, f_α] is some nonzero h_α ∈ 𝔥, equal up to normalisation to the dual Cartan element t_α.

After rescaling, there is a unique sign choice making (e_α, f_α, h_α) satisfy the standard sl₂(ℂ) relations:

So 𝔤_α ⊕ 𝔤_−α ⊕ ℂ·h_α is a Lie subalgebra of 𝔤 isomorphic to sl₂(ℂ), with H ↦ h_α, E ↦ e_α, F ↦ f_α. There is one embedded sl₂ per pair {α, −α}, giving |Δ|/2 sl₂-subalgebras whose union covers every direction in 𝔤 outside the Cartan.

Why this matters. Every semisimple Lie algebra is built out of overlapping copies of a single 3-dimensional algebra. Any fact about sl₂ transports back to 𝔤. The most important transported fact is integrality of eigenvalues.

Integrality, from sl₂ representation theory

The black-box fact about sl₂(ℂ) (proof: a standard undergraduate exercise built from E as raising and F as lowering operators on H-eigenspaces):

Black-box theorem (representations of sl₂). Let V be a finite-dimensional complex representation of sl₂(ℂ). Then V decomposes as a direct sum of H-eigenspaces with strictly integer eigenvalues, and those eigenvalues are symmetric about 0: whenever k is an H-eigenvalue on V, so is −k, with the same eigenspace dimension.

We apply this by making 𝔤 a representation of sl₂(α) via the adjoint action. The basis vectors act as ad_hα, ad_eα, ad_fα, and the theorem says the eigenvalues of ad_hα are integers.

Those eigenvalues are the values of the other roots β on h_α: for y ∈ 𝔤_β, [h_α, y] = β(h_α) · y, so β(h_α) ∈ ℤ for every pair α, β.

Via the iso 𝔥 ≅ 𝔥*, β(h_α) is the Cartan integer of β against α,

(the factor of 2 and denominator come from normalising h_α for integer rather than half-integer eigenvalues). For every ordered pair α, β, the Cartan integer n_β,α is an integer; this is the first hard constraint on Δ.

Weyl reflections

The Cartan integers feed into a reflection. Fix α ∈ Δ and define s_α: 𝔼 → 𝔼 by

Two immediate facts: s_α is the Euclidean reflection across the hyperplane orthogonal to α, since s_α(α) = −α and s_α(ξ) = ξ for ξ ⊥ α; and s_α(β) has integer coefficient −n_β,α on α.

The deeper fact, from the same sl₂ representation theory, is that s_α(β) is itself a root:

So s_α maps Δ to itself. Each root produces a symmetry of Δ, and we can compose them.

Three things to notice about the reflection formula.

(1) It is linear in β. Applying a reflection is a matrix multiply.

(2) It is an involution: applying the same reflection twice returns β to its original position.

(3) If α is a root and β is any root, then s_α(β) is also a root, a property of the root system, not of the formula.

The Weyl group W is the subgroup of the orthogonal group O(𝔼) generated by all the reflections s_α for α ∈ Δ:

W acts on Δ by permutations, embedding W in the symmetric group on |Δ| letters; W is therefore finite. It is the main combinatorial invariant distinguishing simple Lie algebras at the root-system level.

The structural punchline. Every simple Lie algebra over ℂ determines a finite set Δ ⊂ 𝔼 closed under a finite reflection group W. Classifying simple Lie algebras reduces to classifying finite reflection groups on Euclidean space.

Orbits of the Weyl group

Since W permutes the roots, we can ask about its orbits. The orbit of α is the smallest W-stable subset containing α, found by iteratively applying reflections. The Weyl orbits on A₂ (the sl₃ hexagon) and B₂ already show the two characteristic behaviours.

For a simply-laced root system (all roots same squared length), W acts transitively: every root is in one orbit. A₂ and all ADE types are simply-laced. For a non-simply-laced system, W is transitive within each length class: long roots stay long, short stay short. B₂, C₂, B₃, G₂, F₄ are non-simply-laced, with two orbits.

A specific check: the B₂ Cartan integers

Compute the Cartan integers for B₂ directly. The simple roots are α₁ = e₁ − e₂ (long, squared length 2) and α₂ = e₂ (short, squared length 1):

• n_{α₁, α₁} = 2(α₁, α₁)/(α₁, α₁) = 2, same for n_{α₂, α₂}; diagonal entries are always 2.
• (α₁, α₂) = −1, (α₂, α₂) = 1, so n_{α₁, α₂} = 2·(−1)/1 = −2 ∈ ℤ.
• (α₁, α₁) = 2, so n_{α₂, α₁} = 2·(−1)/2 = −1 ∈ ℤ.

The off-diagonal Cartan integers are −2 and −1, integers but unequal. That asymmetry signals a non-simply-laced system: the product n_{α₁, α₂} · n_{α₂, α₁} = 2 controls the angle via n_{α, β} · n_{β, α} = 4 cos²θ. The next explainer treats this as the crystallographic restriction.

One reflection on all 240 of E₈

For E₈ all 240 roots have squared length 2, so s_α(x) = x − ⟨x, α⟩ · α. For any simple root α_i, the reflection fixes the 126 roots perpendicular to α_i and moves the remaining 114.

We can draw each root as a polyline across eight parallel axes. Click a simple root; the polylines recolour, grey for fixed, amber for moved.

The orbit: how 240 comes out of 1

Transitivity is the punchline. Start from α₁, apply simple reflections in any order. For E₈ the orbit grows to all 240 roots: every root is reachable from every other.

The size of the E₈ Weyl group

Every composition of simple reflections (any word in s₁, …, s₈) is also a symmetry of E₈. The full set of compositions is W(E₈).

Every element of the 696-million-element group is a composition of at most a few dozen simple reflections. The formula s_α(β) = β − ⟨β, α⟩α is the only operation; the group emerges from iterating it.

Takeaways

• sl₂ triples. Each root α ∈ Δ determines an embedded sl₂(ℂ) inside 𝔤, spanned by e_α ∈ 𝔤_α, f_α ∈ 𝔤_−α, and h_α ∈ 𝔥 satisfying the sl₂ relations exactly.
• Integrality of eigenvalues (black box). Finite-dimensional sl₂ representations have integer H-eigenvalues symmetric about zero. Applied to 𝔤 as an sl₂(α)-module, this forces every Cartan integer n_{β, α} = 2(β, α)/(α, α) to be an integer.
• Weyl reflections. s_α(β) = β − n_{β, α} · α. An involution, linear, and Δ-preserving. For E₈, one such reflection fixes 126 roots and moves 114.
• The Weyl group. W = ⟨s_α : α ∈ Δ⟩, a finite subgroup of O(𝔼) that permutes Δ. One orbit for simply-laced types, two orbits (long / short) for non-simply-laced. |W(E₈)| = 696,729,600.
• Transitivity. On simply-laced types, a single seed root plus iterated simple reflections reaches every root; the 240 roots of E₈ all come out of one.

Next, the Cartan integers n_{β, α} are constrained beyond integrality: they take values in {0, ±1, ±2, ±3, ±4} because n_{α, β} · n_{β, α} = 4 cos²θ ∈ [0, 4]. That becomes the crystallographic restriction: only four angles between roots are allowed.