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For a Cartan subalgebra $\mathfrak{t}$, a semisimple Lie algebra decomposes asThe nonzero functionals $\alpha$ are the roots.

A root system is a finite set of nonzero vectors closed under the reflections they define and satisfying the crystallographic integrality condition when it arises from a semisimple Lie algebra.

A finite-dimensional Lie algebra is semisimple when it has no nonzero solvable ideals.

A real or complex Lie group is semisimple when its Lie algebra is semisimple. For a real group, the Killing form then defines a nondegenerate bi-invariant pseudo-Riemannian metric. Compact semisimple groups have negative-definite Killing form; noncompact examples such as $SL(2,\mathbb{R})$ have indefinite Killing form. A disconnected group can also have a semisimple Lie algebra, so connectedness is a separate hypothesis when integrating infinitesimal actions.

For a semisimple Lie algebra, its nondegenerate Killing form turns a two-cocycle into an operator $D$ through $B(DX,Y)=\kappa (X,Y)$. The cocycle identity makes $D$ a derivation. Choose $Z$ by $B(Z,X)=\mathrm{tr}(D\mathrm{ad}X)$. Using $[D,\mathrm{ad}X]=\mathrm{ad}(DX)$ and cyclicity of trace gives $B([Z,X],Y)=B(DX,Y)$, so $D=\mathrm{ad}Z$. Consequently $\kappa (X,Y)=B(Z,[X,Y])$ is a coboundary. This proves that component Hamiltonians can be normalized to a Lie algebra homomorphism.

A nonabelian Lie algebra is simple when its only ideals are zero and the entire algebra.

A connected Lie group is simple when its Lie algebra is nonabelian and simple. Some conventions additionally quotient out or exclude a discrete center; the infinitesimal gauge fields depend only on the Lie algebra.

The real traceless two-by-two matrices form a three-dimensional Lie algebra. A basis adapted to conformal mechanics obeys $\{D,E\}=-E$, $\{D,K\}=K$, $\{E,K\}=2D$.

An sl2 triple consists of elements of a Lie algebra satisfying the displayed brackets. It defines a homomorphism from ${\mathfrak{sl}}_2$ by sending its standard generators to $h,e,f$. A nonzero homomorphism is injective because ${\mathfrak{sl}}_2$ is a simple Lie algebra. Root vectors and their coroot give such triples in a semisimple Lie algebra, and their representations supply finite weight strings.

The real Lie algebra of the special unitary group consists of traceless skew-Hermitian matrices. Differentiating unitarity and determinant one gives the two constraints; exponentiating any such matrix proves their sufficiency. The commutator is again skew-Hermitian and traceless. Its real dimension is $n^2-1$, and its complexification is ${\mathfrak{sl}}_n(\mathbb{C})$.

The coefficients $C_{ij}{}^k(q,p)$ in $\{{\phi }_i,{\phi }_j\}=C_{ij}{}^k{\phi }_k$ are structure functions. A finite real span of first-class constraints is a Lie algebra when it closes with constant coefficients. General structure functions instead describe closure with phase-space-dependent coefficients, while still permitting canonical gauge transformations.

The SU(3) group consists of unitary complex three-by-three matrices of determinant one. Its defining action on ${\mathbb{C}}^3$ preserves the Hermitian norm. It contains block-diagonal copies of SU(2) and the real SO(3) group. Its Lie algebra consists of traceless anti-Hermitian matrices.

The compact real Lie algebra $\mathfrak{su}(3)$ consists of traceless anti-Hermitian $3\times 3$ matrices. Physicists often specify Hermitian generators $t_a$ instead, so the genuine Lie algebra elements are $i{t}_a$. The compact real algebra and its complexification ${\mathfrak{sl}}_3(\mathbb{C})$ should be distinguished.

Minimal ten-dimensional supersymmetric gauge theory contains a gauge field and an adjoint Majorana-Weyl spinor. They have eight propagating bosonic and eight propagating fermionic states per gauge generator and sixteen real supercharges. A toroidal zero-mode dimensional reduction to four dimensions gives one gauge field, six real adjoint scalar fields, and four adjoint Weyl spinors: a four-dimensional N=4 super Yang-Mills theory. The internal field strength $F_{mn}=-ig[X_m,X_n]$ gives the nonnegative scalar potential $V=\frac{g^2}{4}{\sum }_{m,n}\Vert [X_m,X_n]{\Vert }^2$, where the norm uses a positive invariant inner product on the Lie algebra. Its minima are commuting tuples of adjoint scalar fields.

For traceless generators, the tensor product of Lie algebra representations has generators $t_a\otimes I+I\otimes s_a$. The trace of their summed squares is the sum of the two factor Casimir traces; the cross term vanishes because $\mathrm{tr}{t}_a=\mathrm{tr}{s}_a=0$. Decomposition into irreducible representations, with multiplicities counted, gives the displayed identity. Without tracelessness an extra term $2{\sum }_a\mathrm{tr}{t}_a\mathrm{tr}{s}_a$ is present. A one-dimensional abelian representation of charge one tensored with itself has Casimir four, not the sum two; semisimple Lie algebras avoid this problem because every generator is a sum of commutators.

An oriented three-dimensional Lie algebra has its antisymmetric structure constants encoded by the displayed matrix, whose symmetric part is $n^{bd}$ and whose antisymmetric part uniquely defines the covector $a_e$. In a volume-normalized basis, the inverse is $C_a{}^b{}_c=\frac{1}{2}{ϵ}_{acd}{n}^{bd}+\frac{1}{2}({\delta }_c^b{a}_a-{\delta }_a^b{a}_c)$. The only independent triple in the Jacobi identity has coefficient $-\frac{1}{2}{n}^{be}{a}_e$, so Jacobi holds exactly when $na=0$. This decomposition is useful in classifying three-dimensional Lie algebras.

Declare a global frame of left-invariant vector fields parallel on a semisimple Lie group. This affine connection is flat and preserves a left-invariant metric tensor because its frame components are constant. If $[e_a,e_c]=C_a{}^b{}_c{e}_b$, its torsion tensor is $T_a{}^b{}_c=-C_a{}^b{}_c$. For fixed $c$, the matrix with entries $T_a{}^b{}_c$ is $\mathrm{ad}{e}_c$. Its trace vanishes because a semisimple Lie algebra is perfect and adjoints of brackets are commutators. The trace of the product for $c,d$ is the Killing form $K_{cd}$, proving both contractions. This flat metric connection differs from the curved torsion-free Levi-Civita connection of the Killing metric.