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A finite-dimensional Lie algebra $L$ is semisimple when its solvable radical is zero, equivalently when it has no nonzero solvable ideals.

We prove the Weyl complete reducibility theorem. Induct on the dimension of a finite-dimensional $L$-module $V$. It is enough first to split a submodule $W$ of codimension one. The one-dimensional quotient is trivial because a semisimple Lie algebra is perfect. By induction, $W$ is a direct sum of irreducible modules. A Casimir element formed using the Killing form commutes with the $L$-action, acts as zero on every trivial summand, and acts by a nonzero scalar on every nontrivial irreducible summand. Its image is therefore the sum $W_1$ of the nontrivial summands, while its kernel contains the trivial summands $W_0$ and maps onto $V/W$. ThusInside $\ker \Omega$, choose a lift $v$ of a basis of $V/W$. For $x\in L$, $xv\in W_0$, and $L$ acts trivially on $W_0$. Hence $[x,y]v=0$ for all $x,y\in L$. Since $L=[L,L]$, actually $xv=0$ for every $x$, so $\mathbb{C}v$ is the required invariant complement.

This codimension-one case implies the general case. For an arbitrary submodule $W\subseteq V$, letwith the natural Hom representation. The maps vanishing on $W$ form an $L$-submodule $X_0$ of codimension one. Splitting $X_0$ supplies an $L$-equivariant $f$ with $f{|}_W=I_W$. Thenso every invariant subspace has an invariant complement and every finite-dimensional representation is completely reducible.

For the requested example, embed ${\mathfrak{sl}}_2$ as the upper-left $2\times 2$ block in the special linear Lie algebra ${\mathfrak{sl}}_3$. Under the restricted Adjoint representation,Here $V_n$ is the irreducible ${\mathfrak{sl}}_2$-module of highest weight $n$. The first summand is the three-dimensional adjoint module, the second is trivial, and the last two are the two-dimensional defining module and its dual, which are isomorphic. This explicit direct sum demonstrates complete reducibility.

A principal sl2 subalgebra of a semisimple Lie algebra has semisimple generator $h_{\mathrm{pr}}=2{\rho }^{\vee }$. Restriction to it packages a representation's weights into ordinary ${\mathfrak{sl}}_2$ weight strings.

The infinitesimal embedding is a subalgebra of the semisimple Lie algebra; integrating it to a Lie-group homomorphism is a separate group-level construction.

The solvable radical of a finite-dimensional Lie algebra is its largest solvable ideal of a Lie algebra. A Lie algebra is semisimple exactly when its solvable radical is zero.

The radical of a bilinear form of the Killing form is the ideal of a Lie algebra $\{x:B(x,y)=0\text{ for all }y\}$. It is zero for a complex semisimple Lie algebra. Every element of the center of a Lie algebra lies in this radical.

The representation ring of finite-dimensional modules for a complex semisimple Lie algebra is the abelian group generated by module classes, with $[V\oplus U]=[V]+[U]$, and multiplication $[V][U]=[V\otimes U]$. Weyl complete reducibility theorem makes the classes of irreducible representations a free integral basis. The formal character map identifies this ring with the Weyl group invariant part of the group ring of a weight lattice. Injectivity follows from distinct highest weights; surjectivity follows by subtracting the character with each maximal dominant weight, whose highest-weight coefficient is one, from a Weyl-invariant finite sum.

// Target: semisimple-lie-algebra.bigb

A root is an individual nonzero vector in a root system. For a complex semisimple Lie algebra, it is a nonzero weight of the Adjoint representation restricted to a Cartan subalgebra; its root space consists of vectors $x$ satisfying $[h,x]=\alpha (h)x$ for all $h$ in that subalgebra.

For a complex semisimple Lie algebra, restrict the module $\mathfrak{h}\oplus {⨁}_{j\ne 0}{\mathfrak{g}}_{j\alpha }$ to the sl2 subalgebra associated with a root. All its weights are even. Its raising operator maps the weight-zero space $\mathfrak{h}$ onto the one-dimensional line of the chosen root vector, so there is exactly one nontrivial irreducible summand: the adjoint module of highest weight $2$. Thus the root spaces at $\pm \alpha$ are one-dimensional and $2\alpha$ is not a root. For any parallel root $c\alpha$, the integers $2c$ and $2/c$ have product $4$; excluding double and half roots leaves only $c=\pm 1$.

For nonproportional roots $\gamma ,\delta$ of a reduced crystallographic root system, $S_{\gamma ,\delta }=\{\delta -p\gamma ,\dots ,\delta +q\gamma \}$ is consecutive, and $p-q=⟨\delta ,{\gamma }^{\vee }⟩$. Realize the crystallographic root system as the roots of a complex semisimple Lie algebra. The sum of the one-dimensional root spaces in this string is a module for the sl2 subalgebra associated with a root. Its weights are $⟨\delta ,{\gamma }^{\vee }⟩+2n$; they have one fixed parity. Complete reducibility gives a sum of irreducible representations of the sl2 Lie algebra. Two summands of this parity would overlap at weight zero when the parity is even, or at weights $\pm 1$ when it is odd, contradicting the one-dimensional root spaces. Thus there is one irreducible string, whose opposite endpoint weights sum to zero. This proves the relation. For distinct simple roots, $p=0$ and the cardinality is $1-⟨\delta ,{\gamma }^{\vee }⟩$. The nonproportional hypothesis excludes $\delta =\pm \gamma$, where zero is not a root and the consecutive-string statement fails.

A root vector is a nonzero element of a root space ${\mathfrak{g}}_{\alpha }$ in the root-space decomposition of a semisimple Lie algebra.

For a Cartan subalgebra $H$ of a finite-dimensional complex semisimple Lie algebra $L$, $H$ is abelian and $C_L(H)=H$, so it is maximal among abelian subalgebras. Starting from the nilpotent self-normalizing definition, the generalized-weight decomposition for a nilpotent Lie algebra and the Engel theorem show that the zero generalized weight space of its adjoint action is exactly $H$. Invariance of the Killing form makes $H$ orthogonal to every nonzero generalized weight space. The Lie theorem gives $B([H,H],H)=0$, and nondegeneracy then forces $[H,H]=0$. An element commuting with $H$ normalizes it, hence belongs to $H$.

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 complex semisimple Lie algebra $\mathfrak{g}\subseteq \mathrm{End}(V)$ and $X\in \mathfrak{g}$, both parts of its Additive Jordan decomposition lie in $\mathfrak{g}$. The polynomial semisimple part of $\mathrm{ad}X$ shows $X_s$ normalizes $\mathfrak{g}$. Split $\mathrm{End}(V)=\mathfrak{g}\oplus M$ by the Weyl complete reducibility theorem; the $M$-component of $X_s$ centralizes $\mathfrak{g}$. On each irreducible summand of $V$ it is scalar by the Schur lemma. Its trace is zero because $\mathfrak{g}$ is a perfect Lie algebra and the nilpotent part has zero trace, so those scalars vanish. Nonsemisimple subalgebras may fail this property: $\mathbb{C}(I+N)$ with $N\ne 0$ nilpotent contains neither Jordan part of $I+N$.

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.

A semisimple Lie algebra over a field of characteristic zero is split if it has a Cartan subalgebra whose adjoint action is simultaneously diagonalizable over that field. Its root-space decomposition then needs no extension of scalars. The sl2R Lie algebra is split, with the real diagonal Cartan subalgebra. The real special unitary Lie algebra $\mathfrak{su}(2)$ is not: its adjoint operators are skew-adjoint for the positive definite inner product obtained by negating its Killing form. Thus a split adjoint operator would have to be zero.

Choosing a positive system of a root system for a complex semisimple Lie algebra divides its root spaces into negative and positive ones. Their respective sums ${\mathfrak{n}}^{-}$ and ${\mathfrak{n}}^{+}$, together with the Cartan subalgebra $\mathfrak{h}$, give this direct sum of vector spaces. Both ${\mathfrak{n}}^{\pm }$ are nilpotent Lie algebras, and $\mathfrak{h}\oplus {\mathfrak{n}}^{+}$ is a Borel subalgebra. The Poincaré-Birkhoff-Witt theorem turns this into an ordered vector-space decomposition of the universal enveloping algebra, useful for building Verma modules.

The trivial representation sends every element of the Lie algebra to the zero linear map. For a complex semisimple Lie algebra, the irreducible representation of highest weight zero is the one-dimensional trivial representation.

A weight-space decomposition expresses a Lie algebra representation as a direct sum of simultaneous eigenspaces for a Cartan subalgebra. Each summand is the weight space for its weight. In complex finite-dimensional representations of a semisimple Lie algebra, the simple coroots act as commuting diagonalizable operators, giving this decomposition.

For a finite-dimensional representation of a complex semisimple Lie algebra, the weights on a line $\mu +\mathbb{Z}\alpha$ form a contiguous string from $\mu +p\alpha$ to $\mu -q\alpha$, with $q-p=⟨\mu ,{\alpha }^{\vee }⟩$. Restriction to the associated sl2 subalgebra gives nested strings of the same parity, so the string endpoints determine the weight set but not its multiplicities.

Every finite-dimensional representation of a complex semisimple Lie algebra is a direct sum of irreducible representations. Equivalently, every invariant subspace has an invariant complement.