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A finite-dimensional complex Lie algebra is semisimple if and only if its Killing form is nondegenerate.

For a finite-dimensional Lie algebra over a field of characteristic zero, Cartan's criterion characterizes a solvable Lie algebra by vanishing of its Killing form on $\mathfrak{g}\times [\mathfrak{g},\mathfrak{g}]$, and a semisimple Lie algebra by nondegeneracy of that form. The Cartan solvability criterion and Cartan criterion for semisimplicity state the two distinct tests.

The Casimir operator formed from the Killing form acts on an Irreducible Lie algebra representation with highest weight $\lambda$ by the scalar $(\lambda ,\lambda +2\rho )$, where $\rho$ is the half-sum of positive roots. It is zero on the trivial Lie algebra representation and nonzero on every nontrivial finite-dimensional irreducible complex representation of a semisimple Lie algebra.

In a finite-dimensional short exact sequence $0\to N\to E\to \mathbb{C}\to 0$ of representations of a complex semisimple Lie algebra, the generalized zero eigenspace of the Casimir operator maps onto the trivial quotient. All its irreducible composition factors have zero Casimir eigenvalue, so they are trivial. The action is therefore strictly upper triangular and has solvable image; because a semisimple Lie algebra is a perfect Lie algebra, that image is zero. Every lift in this generalized zero eigenspace is invariant, giving a split short exact sequence.

In a complex semisimple Lie algebra whose rank of a semisimple Lie algebra is $\ell$, a root vector $x$ satisfies $\dim Z_{\mathfrak{g}}(x)\ge 3\ell -2$. To see this, select $\ell -1$ roots whose images form a basis of the real root span modulo $\mathbb{R}\alpha$, where $x\in {\mathfrak{g}}_{\alpha }$. Take the highest endpoints of the root strings through both signs of each selected root. Their $2(\ell -1)$ distinct root spaces commute with $x$. Together with $\ker \alpha \subset \mathfrak{t}$ and $\mathbb{C}x$, these give $2(\ell -1)+(\ell -1)+1$ independent vectors in the Lie algebra centralizer.

For a complex semisimple Lie algebra and an integer $k\ge 0$, the Weyl character formula and Weyl denominator formula giveApply the dilation $e^{\mu }\mapsto e^{(k+1)\mu }$ to the denominator identity to obtain the numerator. Cancellation leaves a finite geometric series for each positive root of a root system. Evaluating the resulting polynomial at all formal exponentials equal to one gives the dimension, without taking an undefined quotient at the identity.

A Chevalley basis of a complex semisimple Lie algebra consists of suitable coroot elements and one vector $e_{\alpha }$ in each root space, chosen so all Lie bracket structure constants are integers. In particular $[e_{\alpha },e_{\beta }]=N_{\alpha ,\beta }{e}_{\alpha +\beta }$ when the sum is a root, with $N_{\alpha ,\beta }=\pm (p+1)$ where $\beta -p\alpha$ is the last negative-direction root in the root string. For independent roots with no root sum the bracket is zero. The integral lattice makes reduction to other fields possible.

A real Lie algebra is compact when it is the Lie algebra of some compact Lie group. Equivalently, it has a positive-definite invariant bilinear form on a Lie algebra. It is a direct sum of an abelian center and a compact semisimple Lie algebra. For its semisimple part, the compactness criterion from the Killing form is negative-definiteness of the Killing form. This definition concerns the algebra; a global group with that algebra can still have a noncompact central covering direction.

A compact real form of a complex semisimple Lie algebra $\mathfrak{g}$ is a real Lie subalgebra $\mathfrak{u}$ whose complexification of a Lie algebra is $\mathfrak{g}$ and whose Killing form is negative definite. Its simply connected Lie group is compact. The positive internal inner product used in unitary gauge theory is therefore $-\kappa$. For ${\mathfrak{sl}}_2(\mathbb{C})$ the SU(2) Lie algebra is a compact real form, whereas ${\mathfrak{sl}}_2(\mathbb{R})$ is a different real form with indefinite Killing form.

Choose simple root vectors with $[e_i,f_i]=h_i={\alpha }_i^{\vee }$, and let ${\rho }^{\vee }$ be the half-sum of positive coroots. It satisfies ${\alpha }_i({\rho }^{\vee })=1$. Expand $h=2{\rho }^{\vee }={\sum }_i{c}_i{h}_i$. For $i\ne j$, $[e_i,f_j]=0$, since the difference of distinct simple roots is not a root of a root system. Thus $[e,f]=h$, while $[h,e]=2e$ and $[h,f]=-2f$. These identities construct an embedded sl2 Lie algebra in any nonzero complex semisimple Lie algebra. Its raising vector is the principal nilpotent element with a nonzero component in each simple-root space.

A finite root system in a real inner product space is crystallographic when every Cartan integer $2(\beta ,\alpha )/(\alpha ,\alpha )$ is an integer. The roots of a complex semisimple Lie algebra satisfy this because each number is a weight of its sl2 subalgebra associated with a root. Reducedness is a separate axiom; the root-space reducedness lemma establishes it for these Lie-algebra roots.

The derivations of a Lie algebra form a Lie subalgebra of $\mathrm{End}(\mathfrak{g})$ with the commutator bracket. The identity $[D,\mathrm{ad}x]=\mathrm{ad}(Dx)$ makes the inner derivations an ideal of a Lie algebra. For a complex semisimple Lie algebra, the Weyl complete reducibility theorem and the vanishing center of a Lie algebra imply $\mathrm{Der}(\mathfrak{g})=\mathrm{ad}\mathfrak{g}$.

In a finite-dimensional representation of an sl2 triple, start with a nonzero vector $X_0$ satisfying $f{X}_0=0$ and $h{X}_0=-\lambda X_0$, and set $X_n=e^n{X}_0$. The commutator identities give the displayed weight and lowering formulas by induction. If $X_N\ne 0$ and $X_{N+1}=0$, then $f{X}_{N+1}=(N+1)(\lambda -N)X_N$ forces $\lambda =N$, a nonnegative integer. The nonzero terminal vector is essential: choosing a later index after the ladder has already vanished gives no constraint. Applied to the Adjoint representation of a semisimple Lie algebra, this argument proves integrality of Cartan integers.

For a finite-dimensional Irreducible Lie algebra representation of a complex semisimple Lie algebra with highest weight $\lambda$, this recursion computes its weight multiplicities from $m_{\lambda }(\lambda )=1$. Here $\rho$ is the half-sum of positive roots and the inner product is induced by the Killing form. Write the Casimir operator on the weight-$\mu$ space as $(\mu ,\mu +2\rho )I+2{\sum }_{\alpha >0}{F}_{\alpha }{E}_{\alpha }$. The cyclic trace identity between adjacent weight spaces gives $\mathrm{tr}(F_{\alpha }{E}_{\alpha }{|}_{V_{\mu }})={\sum }_{j\ge 1}(\mu +j\alpha ,\alpha )m_{\lambda }(\mu +j\alpha )$. Taking the trace and using the Casimir eigenvalue proves the formula. Only finitely many terms are nonzero.

The group ring of a weight lattice $P$ consists of finite sums ${\sum }_{\lambda }{a}_{\lambda }e(\lambda )$ with integer coefficients, where $e(\lambda )e(\mu )=e(\lambda +\mu )$. A finite-dimensional Lie algebra representation has formal character ${\sum }_{\lambda }(\dim V_{\lambda })e(\lambda )$ in this ring. The Weyl group acts by $e(\lambda )\mapsto e(w\lambda )$. The ambient ring need not consist of genuine characters; the representation ring of a semisimple Lie algebra of finite-dimensional modules for a complex semisimple Lie algebra identifies with the invariant subring $\mathbb{Z}[P{]}^W$.

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For a symplectic left Lie group action, use actual fundamental fields ${\xi }_P$ and ${\iota }_{X_f}\omega =df$. They satisfy $[{\xi }_P,{\eta }_P]=-[\xi ,\eta {]}_P$. Cartan's magic formula then gives ${\iota }_{[\xi ,\eta {]}_P}\omega =d[\omega ({\xi }_P,{\eta }_P)]$. Thus every commutator generator has a globally exact contraction. A semisimple Lie algebra is perfect, so every generator has a global Hamiltonian. The separate semisimple moment-map equivariance result removes the constant two-cocycle obstruction and gives an equivariant moment map for a connected group.

For a finite-dimensional representation of a complex semisimple Lie algebra, select a maximal remaining dominant weight, subtract that multiplicity of the corresponding irreducible formal character, and repeat. The Weyl complete reducibility theorem guarantees nonnegative residual multiplicities and termination. Checking the full character or the exhausted total dimension confirms that no summands remain.

For a complex semisimple Lie algebra and a choice of positive roots, every finite-dimensional Irreducible Lie algebra representation has a unique highest weight, which is a dominant integral weight. Conversely each dominant integral weight labels exactly one such irreducible module up to isomorphism. Its highest-weight vector spans the highest-weight space and generates the module. This classification and Weyl complete reducibility theorem reduce decomposing any finite-dimensional module to its irreducible highest-weight summands.

For a finite-dimensional irreducible representation $L(\lambda )$ of a complex semisimple Lie algebra, its dual Lie algebra representation has highest weight $-w_0\lambda$, where $w_0$ is the longest Weyl-group element. Indeed duality negates all weights, and $w_0\lambda$ is the lowest weight of the original module. In particular $w_0=-I$ makes every such irreducible module self-dual.