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The adjoint representation gave us a linear operator ad_x: 𝔤 → 𝔤 for every x ∈ 𝔤. To understand 𝔤 we want to diagonalise these operators and read off their eigenspaces, the standard manoeuvre for any interesting linear operator.

Each x gives a different ad_x, usually with a different eigenspace decomposition. Rather than pick one, we pick as large a subset as possible where all the adjoint operators can be diagonalised simultaneously. That subset is a Cartan subalgebra, the organising scaffold for everything that follows.

Why we want to diagonalise

An operator T on V is diagonalisable if V decomposes as a direct sum of T's eigenspaces. That decomposition is the cleanest description of T: which subspaces are invariant, what T does on each, and how to take functions of T (powers, exponentials, logarithms) eigenspace-by-eigenspace. Given a Lie algebra 𝔤 whose adjoint operators are simultaneously diagonalisable on some 𝔥 ⊂ 𝔤, we get joint eigenspaces indexed by linear functionals 𝔥 → ℂ. The structure those functionals are forced into is the root system.

Simultaneous diagonalisation requires two conditions: each operator must be diagonalisable, and any two must commute. Both are facts about linear algebra on a fixed vector space:

Linear-algebra fact. A family of linear operators {T_i} on a finite-dimensional vector space V can be simultaneously diagonalised, meaning there is a single basis of V that is an eigenbasis for every T_i at once, if and only if (a) each T_i is diagonalisable on its own, and (b) T_i T_j = T_j T_i for all i, j. These two conditions are necessary and sufficient.

So we need a subspace 𝔥 ⊂ 𝔤 of elements x for which ad_x is diagonalisable and any two give commuting adjoint operators. The second condition has a clean translation: ad_x and ad_y commute iff [x, y] lies in the centre of 𝔤. For simple Lie algebras the centre is zero, so the condition becomes [x, y] = 0. We are looking for a subspace of commuting elements, each with a diagonalisable adjoint.

Toral subalgebras and the Cartan

An element x ∈ 𝔤 whose adjoint ad_x is diagonalisable is called semisimple. A subspace 𝔥 ⊂ 𝔤 is toral if every element is semisimple and 𝔥 is abelian. A Cartan subalgebra is a maximal toral subalgebra.

"Maximal" means: you cannot extend 𝔥 by any new semisimple y ∉ 𝔥 while keeping it abelian. So a Cartan subalgebra captures the largest possible commuting diagonalisable family.

Three theorems about Cartan subalgebras hold for every finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero (such as ℂ, the field we use throughout). We use each one as a black box:

1. Existence. A Cartan subalgebra exists. One classical construction picks any element whose adjoint is as close to being "generic" as possible and takes the centraliser of its semisimple part.
2. Abelianness in the simple case. For a simple (or more generally semisimple) 𝔤, the Cartan subalgebra 𝔥 is honestly abelian as a subspace of 𝔤; there is no hidden nilpotent part. This is what lets us treat ad_h for h ∈ 𝔥 as a family of genuinely commuting diagonalisable operators.
3. Conjugacy. Any two Cartan subalgebras are conjugate to each other under an inner automorphism of 𝔤. Equivalently, the abstract data we extract from 𝔥 — its dimension, its eigenspace structure on 𝔤, and the geometry of the resulting root system — does not depend on which Cartan subalgebra we picked.

Conjugacy is the one we lean on most. The dimension of any Cartan subalgebra is an invariant of 𝔤 itself.

Rank. The rank of a semisimple Lie algebra is the dimension of any Cartan subalgebra. The nine families in the classification are indexed by rank: A₁, A₂, A₃, … for sl_n; E₆, E₇, E₈ for the rank-6, 7, 8 exceptionals; and so on.

Diagonal matrices in sl_n

The construction is easiest in sl_n(ℂ). The diagonal trace-zero matrices form an (n − 1)-dimensional abelian subalgebra (any two diagonal matrices commute). Each is trivially diagonalisable, so each adjoint operator is too. The subspace is maximal: you cannot enlarge it while keeping it abelian and semisimple. So it is a Cartan subalgebra of sl_n, and the rank of sl_n is n − 1.

Concretely, for sl₃ the Cartan subalgebra is the 2-dimensional space of diagonal matrices

where h₁ + h₂ + h₃ = 0 kills one degree of freedom. sl₃ is 8-dimensional. Two of those dimensions are the diagonal Cartan; the other six are the matrix units E_ij (with i ≠ j). For H = diag(h₁, h₂, h₃) the adjoint gives [H, E_ij] = (h_i − h_j) E_ij, so E_ij is an eigenvector with eigenvalue functional H ↦ h_i − h_j.

The joint eigenspace decomposition

Pick any Cartan subalgebra 𝔥 ⊂ 𝔤. Because 𝔥 is abelian and semisimple, {ad_h: h ∈ 𝔥} is a commuting family of diagonalisable operators on 𝔤, and the family can be simultaneously diagonalised. So 𝔤 decomposes as a direct sum of joint eigenspaces: for each linear functional α: 𝔥 → ℂ, define

Each 𝔤_α is the joint eigenspace with joint eigenvalue α. Only finitely many α give nontrivial eigenspaces; these are the roots of 𝔤 with respect to 𝔥. The root system is

the set of nonzero functionals with non-empty eigenspaces. Including α = 0 (where 𝔤₀ = 𝔥), the direct sum gives the root space decomposition:

Every element of 𝔤 is uniquely a Cartan element plus one contribution from each root space. Four structural facts about this decomposition drive the classification:

The four facts.

(1) 𝔤₀ = 𝔥. The zero eigenspace equals the Cartan subalgebra itself. Nothing else has joint eigenvalue zero.

(2) Root spaces come in pairs. If α ∈ Δ, then −α ∈ Δ. The roots of a semisimple Lie algebra are symmetric about the origin in 𝔥*.

(3) Each root space is one-dimensional. For every α ∈ Δ, dim 𝔤_α = 1. In a simple Lie algebra, each root α identifies exactly one direction in 𝔤, up to scalar.

(4) Brackets are additive on roots. If x ∈ 𝔤_α and y ∈ 𝔤_β, then [x, y] ∈ 𝔤_α+β. In particular, [𝔤_α, 𝔤_β] is zero unless α + β is zero or a root.

Facts (1), (2), and (4) follow from joint-eigenspace arguments and the Jacobi identity. Fact (3) is the deep one, holding specifically for semisimple Lie algebras; its proof uses the embedded sl₂ triples of the next explainer. It is the most restrictive piece of structure here and what makes the classification possible.

sl₃ as the concrete instance

All four facts can be verified directly in sl₃(ℂ). The Cartan is the 2-dimensional subspace above; the other six dimensions are spanned by the matrix units E_ij (1 in position (i, j), 0 elsewhere, i ≠ j). For H = diag(h₁, h₂, h₃) ∈ 𝔥,

So E_ij is a joint eigenvector for every H ∈ 𝔥, with eigenvalue α_ij(H) = h_i − h_j. The functional α_ij is a root, and span(E_ij) is the root space. The six pairs (i, j) with i ≠ j give six roots. Writing ε_i(H) = h_i,

which is the root system A₂. Six roots in a 2-dimensional space 𝔥*, sitting at the vertices of a regular hexagon.

Fact (1) says nothing else in sl₃ is killed by all ad_H. Fact (2) is immediate: swapping (i, j) negates h_i − h_j. Fact (3) holds by inspection. For fact (4):

The functional (h_i − h_j) + (h_j − h_k) = h_i − h_k is α_ik, so [E_ij, E_jk] lands in 𝔤_αᵢₖ. When the j's fail to match, the product is zero: the matrix-unit version of "α + β is not a root, so the bracket vanishes."

Why one-dimensionality matters

Fact (3) is the assertion worth pausing on: every nonzero root space is one-dimensional. This turns an infinite-dimensional classification problem into a finite combinatorial one.

If 𝔤 has rank r and dimension d, the Cartan takes r of those dimensions and the remaining d − r are distributed over the root spaces. If each 𝔤_α had arbitrary dimension, we would need to record those dimensions separately. One-dimensionality gives |Δ| = d − r: once we know the root system Δ in 𝔥*, we know everything about 𝔤 beyond the Cartan up to isomorphism.

This is where "classify semisimple Lie algebras" becomes "classify finite configurations of vectors."

Rank counting. For sl_n+1: dimension n(n+2), rank n, so |Δ| = n(n+2) − n = n(n+1). That's the root count of A_n. For so(2n+1): dimension n(2n+1), rank n, so |Δ| = 2n². For E₈: dimension 248, rank 8, so |Δ| = 240. Every one of the nine families in the classification satisfies this arithmetic, and the arithmetic works because root spaces are one-dimensional.

Root space decomposition for the classical families

The sl₃ example generalises to every classical family the same way: the Cartan is a space of diagonal (or block-diagonal) matrices, and the root spaces are spanned by off-diagonal matrix units.

Type	Lie algebra	Cartan 𝔥	Roots	|Δ|
An	sln+1(ℂ)	diagonal, trace 0	εi − εj, i ≠ j	n(n+1)
Bn	so(2n+1, ℂ)	diagonal, n entries	±εi ± εj, ±εi	2n²
Cn	sp(2n, ℂ)	diagonal, n entries	±εi ± εj, ±2εi	2n²
Dn	so(2n, ℂ)	diagonal, n entries	±εi ± εj	2n(n−1)

For the exceptional Lie algebras G₂, F₄, E₆, E₇, E₈, no matrix realisation falls out cleanly, so we build the Cartan abstractly: write down the full root system from combinatorial data (Dynkin diagram + Cartan matrix) and read the Cartan off as the span of the simple co-roots.

Bracket additivity and the root string

Bracket additivity is the workhorse of every structural calculation. More explicitly:

In a simple Lie algebra [𝔤_α, 𝔤_β] = 𝔤_α+β whenever α + β is a root, with no hidden cancellation. The bracket surjects onto its target. So for any root α and any β ∈ Δ, the sequence

walking from β in the direction of α, is an unbroken consecutive run p ≤ k ≤ q with no gaps. This is the α-string through β, and its length p + q + 1 is one of the integer constraints the classification extracts. Root strings have at most four elements, which forces the crystallographic restriction on root-system angles.

The Killing form

We have Δ as a finite set of linear functionals on 𝔥. To classify it we need geometry: when are two roots close, what angle do they form, are they long or short?

This section builds, from nothing more than the bracket, a canonical bilinear form on 𝔤, the Killing form; restricts it to 𝔥; transports it to 𝔥* via 𝔥 ≅ 𝔥*; and verifies it is positive definite on the real span of the roots. At that point the roots become vectors in a real Euclidean space 𝔼.

Composing two adjoint operators ad_x ∘ ad_y gives a linear operator on 𝔤 whose trace is a coordinate-independent scalar. That trace defines the Killing form:

This is a bilinear form κ: 𝔤 × 𝔤 → ℂ, symmetric by cyclicity of the trace. It has one further property, ad-invariance:

which says ad_z acts as a "skew derivation" with respect to κ. Ad-invariance reflects the Jacobi identity and makes κ the only natural bilinear form on 𝔤 up to scaling (on a simple 𝔤).

To compute κ, pick a basis of 𝔤, write each ad_x as a matrix, multiply, and trace. For sl₂(ℂ) in the (H, E, F) basis,

and the resulting Killing form on sl₂, expressed as the 3×3 matrix of values κ(basis_i, basis_j), is:

For sl_n(ℂ) the Killing form has a closed form bypassing the basis calculation:

Thus in sl_n, the Killing form is, up to 2n, the trace of a matrix product. κ(X, X) = 2n · tr(X²) is strictly positive on any nonzero real diagonal X, which is the toehold for proving positive-definiteness on the real span of the roots.

Cartan's criterion

Non-degeneracy of the Killing form is the defining property of semisimple Lie algebras (Cartan, taken as a black box):

Cartan's criterion. A finite-dimensional Lie algebra 𝔤 over a field of characteristic zero is semisimple if and only if its Killing form is non-degenerate: for every nonzero x ∈ 𝔤 there is some y ∈ 𝔤 with κ(x, y) ≠ 0. Equivalently, the Gram matrix of κ in any basis has nonzero determinant.

For sl_n non-degeneracy is verified by computation. For an abstract semisimple algebra, non-degeneracy is how one knows it is semisimple. The criterion bridges "no proper nontrivial ideals" and "non-degenerate bilinear form."

Non-degeneracy on 𝔤 implies non-degeneracy on the Cartan 𝔥. The restriction κ|_{𝔥 × 𝔥} gives an inner-product structure on 𝔥 that pushes forward to 𝔥*, and that push-forward is where the roots become vectors.

From 𝔥 to 𝔥* via the Killing form

A non-degenerate bilinear form on V gives a canonical isomorphism V ≅ V*: x ↦ (y ↦ form(x, y)). For the Killing form on 𝔥, this is

Given a root α ∈ 𝔥*, the inverse iso produces a unique t_α ∈ 𝔥 with

t_α is the dual Cartan element of α: pairing with t_α via κ equals applying α, the standard raise/lower-index passage with κ as metric.

Transporting κ from 𝔥 to 𝔥* gives the induced inner product

a non-degenerate symmetric bilinear form on 𝔥*. Now for every root α we can ask whether (α, α) is positive.

Positive-definiteness on the real span

The fact that anchors the whole classification:

Theorem. The induced inner product (·, ·) on the real linear span of the roots is strictly positive definite: for any nonzero ξ = Σ c_α α in the real span of Δ, (ξ, ξ) > 0.

The proof uses three ingredients. The Killing form takes integer values on the co-root lattice. For any root α, κ(t_α, t_α) is a positive integer (a weighted count of the α-string). The Gram matrix of any independent set of roots is therefore symmetric with positive diagonal and integer off-diagonals, which is positive definite by a linear-algebra lemma.

So Δ lives inside a real vector space

of dimension equal to the rank of 𝔤, with an honest Euclidean inner product. Distances, angles, and lengths between roots are well-defined. From here every classification argument is a statement about the geometry of a finite configuration of vectors in 𝔼.

Killing form on the sl₃ Cartan

The simplest nontrivial instance is sl₃(ℂ). Pick a basis

and use κ(X, Y) = 6 tr(XY) on sl₃. Direct calculation gives

The Killing form matrix on the Cartan, in this basis, has strictly positive eigenvalues, confirming 𝔥 is a rank-2 Euclidean space.

What the shift to geometry buys us

Once we are doing Euclidean geometry on Δ ⊂ 𝔼, a lot of machinery becomes available. Three consequences we will cash in on from the next explainer onward:

• Angles and lengths. Every root has a squared length (α, α) > 0, and every pair of roots has a well-defined angle via the standard cosine formula. In the classification, only a handful of angles are allowed (90°, 120°, 135°, 150°) and the discreteness of the allowed set is what makes the classification tractable.
• Reflections. Each root α defines a reflection of 𝔼 across the hyperplane perpendicular to α. Because these reflections send roots to roots (this is the sl₂ fact from the next explainer), they generate a finite reflection group, the Weyl group, and that group is the second main actor of the classification story.
• Finite configurations of vectors. The root system Δ is now a finite, symmetric-about-the-origin configuration of vectors in 𝔼, and we can ask the combinatorial question: which such configurations are allowed? The answer, it turns out, is spectacularly restrictive.

The pivot. Until now we have been doing algebra; from here on we do Euclidean geometry. The bridge is κ, and the positive-definiteness theorem is why it holds up.

Takeaways

• Cartan subalgebra. A maximal abelian subspace of 𝔤 all of whose elements are semisimple. Rank = dim 𝔥 is an invariant of 𝔤 itself.
• Root space decomposition. 𝔤 = 𝔥 ⊕ ⊕_{α ∈ Δ} 𝔤_α, with each root space one-dimensional, roots paired α ↔ −α, and brackets additive on roots.
• One-dimensionality. Turns "classify Lie algebras" into "classify finite vector configurations." |Δ| = dim 𝔤 − rank 𝔤, cleanly explaining why E₈ (dim 248, rank 8) has exactly 240 roots.
• Killing form. κ(x, y) = tr(ad_x ad_y), a canonical ad-invariant symmetric bilinear form. Non-degenerate iff 𝔤 is semisimple (Cartan's criterion).
• Geometric root system. κ restricts non-degenerately to 𝔥, pushes forward to 𝔥* via 𝔥 ≅ 𝔥*, and is strictly positive definite on 𝔼 = span_ℝ Δ. The roots are now honest vectors in a Euclidean space, with well-defined lengths and angles.

Next we use sl₂ triples embedded in 𝔤 to extract the first restrictive constraint on Δ: Weyl reflections send roots to roots, forcing the pairing ⟨α, β⟩ := 2(α, β)/(β, β) to be an integer for every pair of roots. The Weyl group emerges from that integrality.