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

The bracket

Lie algebras arise as tangent spaces at the identity of Lie groups, groups that are also smooth manifolds, such as the rotation group SO(3); the Lie bracket records how two infinitesimal motions fail to commute.

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.

The prototype is sl₂(ℂ), the 2×2 complex matrices with trace zero, with the commutator [X, Y] = XY − YX as bracket. It is spanned by

with structure constants

These relations fix every bracket in sl₂, and Part 2 finds a copy of them inside every simple Lie algebra.

sl_n(ℂ), the trace-zero n×n matrices, has dimension n² − 1 and is type A_n−1. The skew-symmetric matrices so(n) are the tangent space of the rotation group. For n = 3 the general element is

and a direct check gives

so so(3) is ℝ³ with the cross product.

Simple algebras and the adjoint action

An ideal is a subspace 𝔦 with [𝔤, 𝔦] ⊆ 𝔦, the analogue of a normal subgroup. A Lie algebra is simple if it is non-abelian and its only ideals are {0} and 𝔤; a semisimple algebra is a direct sum of simple ones.

The classification runs on the adjoint representation. Each x ∈ 𝔤 acts 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

Commuting diagonalisable operators can be diagonalised in one basis. Call x semisimple when ad_x is diagonalisable. A Cartan subalgebra 𝔥 is a maximal abelian subalgebra of semisimple elements. Any two are conjugate by an automorphism of 𝔤, so their dimension, the rank, is an invariant.

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, with an eigenvalue that is a linear functional on 𝔥: α_ij(H) = h_i − h_j.

Roots

In any semisimple 𝔤, collect for each linear functional α on 𝔥 the joint eigenvectors:

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

Four facts carry the classification. The zero eigenspace is 𝔥 itself. If α is a root, so is −α. [𝔤_α, 𝔤_β] ⊆ 𝔤_α+β, which vanishes unless α + β is zero or a root. Each root space is one-dimensional. The third is the Jacobi identity, the first two need the Killing form below, and the fourth needs the sl₂ subalgebras of Part 2.

For sl₃ the six roots are

and additivity is multiplication of matrix units: [E_ij, E_jk] = E_ik for i ≠ k, while a product whose inner indices fail to match is zero.

Elsewhere the roots have to be solved for. Figure 3 does this for three classical algebras of rank 2 and for the derivations of the octonions, which Part 6 identifies as G₂.

Since each root space is one-dimensional, |Δ| = dim 𝔤 − rank: n(n + 1) for sl_n+1, and 248 − 8 = 240 for E₈. 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 Killing form

Lengths and angles come from 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 a simple algebra every invariant symmetric form is a multiple of κ, so the trace form tr(XY) of a matrix model is one too. On sl_n,

Non-degeneracy survives restriction to 𝔥, so 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. For h ∈ 𝔥, ad_h is diagonal, with eigenvalue 0 on 𝔥 and α(h) on each 𝔤_α, so

and for λ in the span of the roots

Part 2 shows that 2(β, α)/(α, α) is an integer. Dividing the identity for λ = α by (α, α)² gives 1/(α, α) = ¼ Σ_β (2(β, α)/(α, α))², so every (α, β) is rational, and (λ, λ) is a sum of rational squares that vanishes only if λ is orthogonal to every root, that is, λ = 0. So the roots live in a Euclidean space of dimension equal to the rank,

Each root now has a reflection in the hyperplane perpendicular to it. Part 2 proves the integrality used above and shows that these reflections permute the roots and generate a finite group.