nLab Killing form



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The Killing form or Cartan-Killing form is a binary invariant polynomial that is present on any finite-dimensional Lie algebra.


Given a finite-dimensional kk-Lie algebra 𝔤\mathfrak{g} its Killing form B:𝔤𝔤kB:\mathfrak{g}\otimes \mathfrak{g}\to k is the symmetric bilinear form given by the formula

B(x,y)=tr(ad(x)ad(y)), B(x,y) \,=\, tr\big( ad(x) \circ ad(y) \big) \,,


ad(x)[x,]:𝔤𝔤 ad(x) \,\coloneqq\, [x,-] \,:\, \mathfrak{g} \longrightarrow \mathfrak{g}

is the linear map given by the adjoint action of xx, hence the value on xx of the adjoint representation ad:𝔤Der(𝔤)ad \colon \mathfrak{g} \to Der(\mathfrak{g}).

If {t a}\{t_a\} is a linear basis for 𝔤\mathfrak{g} and {C a bc}\{C^a{}_{b c}\} are the structure constants of the Lie algebra in this basis (defined by [t a,t b]= cC ab ct c[t_a, t_b] = \sum_c C^c_{a b} t_c), then

B(t a,t b)= c,dC c adC bc d. B(t_a, t_b) \,=\, \sum_{c,d} C^c{}_{a d} C^{d}_{b c} \,.


The Killing form is am invariant polynomial in that

B([x,y],z)=B(x,[y,z]) B\big([x,y],z\big) \,=\, B\big(x, [y,z] \big)

for all x,y,z𝕘x,y,z \in \mathbb{g}. This follows from the cyclic invariance of the trace],

For complex Lie algebras 𝔤\mathfrak{g}, nondegeneracy of the Killing form (i.e. being the metric making 𝔤\mathfrak{g} a metric Lie algebra) is equivalent to semisimplicity of 𝔤\mathfrak{g}.

For simple complex Lie algebras, or compact forms of real Lie algebras, any invariant nondegenerate symmetric bilinear form is proportional to the Killing form. Indeed, one often uses a normalisation of the Killing form on a simple Lie algebra that makes it a positive definite inner product, such that the algebra elements corresponding to long roots (under the isomorphism 𝔤 *𝔤\mathfrak{g}^*\simeq \mathfrak{g} induced by the original Killing form) have length 2\sqrt{2}. Both the Killing form and this normalised version give a simple Lie algebra a canonical structure of a metric Lie algebra, one positive definite, and one negative definite. This normalised Killing form is sometimes called, following Pressley and Segal, the basic inner product on the Lie algebra.

Here are the basic inner products for the compact forms of the simple real matrix Lie algebras, from (Wang–Ziller 1985, page 583):

  • 𝔰𝔲(n)\mathfrak{su}(n): X,Y=tr (XY)\langle X,Y\rangle = -tr_{\mathbb{C}}(XY).
  • 𝔰𝔬(3)\mathfrak{so}(3): X,Y=tr (XY)/4\langle X,Y\rangle = -tr_{\mathbb{R}}(XY)/4.
  • 𝔰𝔬(n)\mathfrak{so}(n), n5n\geq 5: X,Y=tr (XY)/2\langle X,Y\rangle = -tr_{\mathbb{R}}(XY)/2.
  • 𝔰𝔭(n)\mathfrak{sp}(n): X,Y=Tr(XY)=2tr (XY)\langle X,Y\rangle = -Tr(XY) = -2\Re tr_{\mathbb{H}}(XY).

The trace tr k()tr_{k}(\cdot) here is the ordinary trace for matrices over the division ring kk, and the reduced trace Tr()Tr(\cdot) for n×nn\times n quaternionic matrices is the composite of the embedding into 2n×2n2n\times 2n complex matrices (thinking of j\mathbb{H}\simeq \mathbb{C}\oplus j\mathbb{C}), and the ordinary trace of complex matrices.


Sometimes one considers more generally a Killing form B ρB_\rho for a more general faithful finite-dimensional representation ρ\rho, B ρ(x,y)=tr(ρ(x)ρ(y))B_\rho(x,y) = tr\big(\rho(x)\rho(y)\big). If the Killing form is nondegenerate and x 1,,x nx_1,\ldots,x_n is a basis in LL with x 1 *,,x n *x_1^*,\ldots,x_n^* the dual basis of 𝔤 *\mathfrak{g}^*, with respect to the Killing form for ρ\rho, then the canonical element r= ix ix i *r = \sum_i x_i\otimes x_i^* defines the Casimir operator C(ρ)=(ρρ)(r)C(\rho) =(\rho\otimes\rho)(r) in the representation ρ\rho; regarding that the representation is faithful, if the ground field is \mathbb{C}, by Schur's lemma C(ρ)C(\rho) is a nonzero scalar operator. Instead of Casimir operators in particular faithful representations it is often useful to consider an analogous construction within the universal enveloping algebra, the Casimir element in U(𝔤)U(\mathfrak{g}).


For a general discussion see:

Careful discussion around normalisation and related data is in:

  • McKenzie Y. Wang and Wolfgang Ziller, On normal homogeneous Einstein manifolds, Annales scientifiques de l’École Normale Supérieure, Série 4 18 (1985), no. 4, 563–633.

Last revised on May 7, 2022 at 02:35:47. See the history of this page for a list of all contributions to it.