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By the Cartan–Killing theorem, every finite-dimensional simple Lie algebra over ℂ is, up to isomorphism, in one of four infinite families, the classical types A, B, C, D, or one of five exceptions: G₂, F₄, E₆, E₇, E₈. The first step of the proof, and the subject of this part, turns the bracket of a Lie algebra into a finite set of vectors whose geometry can be classified.

The bracket

Lie algebras come from Lie groups, groups that are also smooth manifolds, like the rotation group SO(3). Near the identity a Lie group looks like its tangent space. For SO(3) that space is 3-dimensional: infinitesimal rotations, each an axis with an angular speed. The vector space alone forgets most of the group. The Lie bracket records the second-order failure of two infinitesimal motions to commute, and that is enough to rebuild the group near the identity.

Abstractly, a Lie algebra is a vector space 𝔤 (over ℂ throughout) with a bracket [·, ·]: 𝔤 × 𝔤 → 𝔤 that obeys three laws:

1. Bilinearity.
2. Antisymmetry.
3. Jacobi identity.

The bracket is not associative; the Jacobi identity takes its place. Bilinearity and antisymmetry give [x, x] = 0 for every x.

The prototype is sl₂(ℂ), the 2×2 complex matrices with trace zero, with bracket the commutator [X, Y] = XY − YX. The commutator of any two matrices has trace zero because tr(XY) = tr(YX), so the bracket stays inside sl₂. It is three-dimensional, spanned by

with structure constants

By bilinearity these three relations determine every bracket in sl₂. They return in Part 2, where every root of every simple Lie algebra carries its own copy of sl₂.

Two more families appear everywhere. sl_n(ℂ), the trace-zero n×n matrices, has dimension n² − 1 and is the type A_n−1 of the classification. so(n), the skew-symmetric matrices, is the tangent space of the rotation group. For n = 3 its general element is

and a direct check gives

so so(3) and ℝ³ with the cross product are the same Lie algebra.

Simple algebras and the adjoint action

An ideal 𝔦 ⊂ 𝔤 is a subspace with [x, y] ∈ 𝔦 for every x ∈ 𝔤 and y ∈ 𝔦. Ideals play the role of normal subgroups: 𝔤/𝔦 inherits a bracket exactly when 𝔦 is an ideal. A Lie algebra is simple if it is non-abelian and its only ideals are {0} and 𝔤. Every finite-dimensional semisimple Lie algebra over ℂ is a direct sum of simple ones, so classifying the simple ones is enough.

The classification runs on the adjoint representation. Each x ∈ 𝔤 gives a linear map on 𝔤 itself,

and the Jacobi identity says exactly that x ↦ ad_x preserves brackets:

So every element of 𝔤 is also a matrix acting on 𝔤, with eigenvalues and eigenspaces.

The Cartan subalgebra

Different elements give different eigenspace decompositions, so we diagonalise as many at once as possible. Operators can be diagonalised in one basis exactly when each is diagonalisable and they pairwise commute. Call x semisimple when ad_x is diagonalisable. A Cartan subalgebra 𝔥 is a maximal abelian subalgebra of semisimple elements. Any two Cartan subalgebras are conjugate by an automorphism of 𝔤, so their common dimension, the rank, is an invariant of 𝔤.

In sl_n(ℂ) the diagonal trace-zero matrices form a Cartan subalgebra, so sl_n has rank n − 1. For sl₃:

For H = diag(h₁, h₂, h₃) and a matrix unit E_ij (a 1 in position (i, j), zero elsewhere),

so every E_ij with i ≠ j is an eigenvector of every ad_H at once. Its eigenvalue depends linearly on H, which makes it a linear functional on 𝔥: α_ij(H) = h_i − h_j.

Roots

The same thing happens in any semisimple 𝔤. For each linear functional α on 𝔥, collect the joint eigenvectors with eigenvalue α:

The nonzero α with 𝔤_α ≠ 0 are the roots, a finite set Δ ⊂ 𝔥*, and 𝔤 splits as

Four facts about this decomposition carry the classification. The zero eigenspace is 𝔥 itself. If α is a root, so is −α. Brackets add roots: [𝔤_α, 𝔤_β] ⊆ 𝔤_α+β, which vanishes unless α + β is zero or a root. Each root space is one-dimensional. The first three follow from the Jacobi identity and joint-eigenspace bookkeeping. The fourth, which makes the classification finite, needs the sl₂ subalgebras of Part 2.

For sl₃ all four can be read off the matrix. The six roots are

and bracket additivity is matrix multiplication of units: [E_ij, E_jk] = E_ik for i ≠ k, the functional (h_i − h_j) + (h_j − h_k) = h_i − h_k. When the inner indices fail to match, the product is zero, the matrix version of "α + β is not a root."

One-dimensionality turns a question about algebras into one about finite sets of vectors. If 𝔤 has dimension d and rank r, the root spaces fill the other d − r dimensions one each, so |Δ| = d − r. For sl_n+1 that is n(n + 2) − n = n(n + 1). For E₈, dimension 248 and rank 8, it is 240. Knowing Δ determines 𝔤 up to isomorphism.

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

The exceptional algebras have no equally clean matrix model, so for them one builds the root system first, from the combinatorial data of Part 3, and the algebra from it.

The Killing form

To classify root systems we need lengths and angles, and the bracket supplies a canonical bilinear form, the Killing form:

It is symmetric, because trace is cyclic, and invariant under the adjoint action:

Cartan's criterion says a Lie algebra over ℂ is semisimple exactly when κ is non-degenerate. On sl_n there is a closed form,

Non-degeneracy survives restriction to 𝔥, so κ identifies 𝔥 with its dual. Each root α has a partner t_α ∈ 𝔥 defined by

and κ moves over to an inner product on roots:

The Euclidean space of roots

On the real span of the roots, this inner product is positive definite, and the rest of the series depends on that. To see it, evaluate κ on 𝔥 using the root decomposition. For h ∈ 𝔥, ad_h is diagonal, with eigenvalue 0 on 𝔥 and α(h) on each one-dimensional 𝔤_α. So the trace of ad_h ad_h′ is a sum over roots:

Applied to t_λ for λ in the span of the roots, this reads

The sl₂ analysis of Part 2 shows that 2(β, α)/(α, α) is an integer for every pair of roots. Dividing the identity above for λ = α by (α, α)² gives 1/(α, α) = ¼ Σ_β (2(β, α)/(α, α))², a positive rational. So every (α, α) is rational, and then every (α, β) is too. On the rational span of the roots, (λ, λ) is a sum of squares of rationals. It vanishes only if λ is orthogonal to every root, and non-degeneracy then forces λ = 0. Choosing a basis of roots, the Gram matrix is rational and positive definite, so it stays positive definite on the real span.

The roots therefore live in a real Euclidean space of dimension equal to the rank,

and from here on every argument is about the geometry of a finite set of vectors in 𝔼.

Once the roots are vectors, each root has a reflection in the hyperplane perpendicular to it. Part 2 shows that these reflections permute the roots, proves the integrality used above, and assembles the reflections into a finite group, the Weyl group.