The Killing form on a $n-$dimensional Lie algebra $\mathrm{\𝔤}$ is the symmetric quadratic form $B$defined by $B\left(x\, y\right) \=$trace$\left(\mathrm{ad}\left(x\right)\cdot \mathrm{ad}\left(y\right)\right)$for any $x\, y \in \mathrm{\𝔤}$. Here $\mathrm{ad}\left(x\right)$ and $\mathrm{ad}\left(y\right)$ are the adjoint matrices for the vectors $x$ and $y\.$In terms of the structure constants $C_{\mathrm{ij}}^k$with respect to the basis {$e_{i }$}for $\mathrm{\𝔤}\mathit{\,}$one has $b_{\mathrm{ij}} \= B\left(e_i\, e_j\right) \=\sum _{k\,\mathrm{\ℓ} \=1}^n{C}_{i\mathrm{\ℓ}}^k C_{\mathrm{kj}}^{\mathrm{\ℓ}}$. If $\mathrm{\𝔥}$$\subset \mathrm{\𝔤}$is a subspace with basis $\left\{x_1\, x_2\, ..\. \,x_p\right\}$, then the restriction of the Killing form to $𝔥$ is given by the $p \times p$matrix $\overline{b_{}}{ }_{\mathrm{rs}} \= B\left(x_r\, x_s\right)\.$

 Killing() returns the $n \times n$symmetric matrix $\left[b_{\mathrm{ij}}\right]$ for the Lie algebra defined by the current frame. Killing(Alg) returns the $n \times n$symmetric matrix $\left[b_{\mathrm{ij}}\right]$ for the Lie algebra Alg. Alg.Killing(h) returns the Killing Matrix $\left[\overline{b_{}}{ }_{\mathrm{rs}}\right]$restricted to the subalgebra $\mathrm{\𝔥}\mathit{\.}$

KillingForm(Alg) returns the symmetric rank 2-tensor $b_{\mathrm{ij} }{\mathrm{\θ}}^i \otimes {\mathrm{\θ}}^j$, where the {${\mathrm{\θ}}^i\}$are the dual 1-forms to the basis {$e_{i }\}$.

The command Killing is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form Killing(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Killing(...).

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{LieAlgebras}\right)\:$

$\mathrm{L1}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg1}\,\left[3\right]\right]\,\left[\left[\left[2\,3\,1\right]\,1\right]\,\left[\left[1\,3\,2\right]\,-1\right]\,\left[\left[1\,2\,3\right]\,1\right]\right]\right]\right)\:$

$\mathrm{DGsetup}\left(\mathrm{L1}\right)\:$

$\mathrm{MultiplicationTable}\left(''LieBracket''\right)$

$\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=\mathrm{e3}\,\left[\mathrm{e1}\,\mathrm{e3}\right]\=-\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\right]$

$X≔\mathrm{evalDG}\left(\mathrm{e1}+\mathrm{e3}\right)$

$X:=\mathrm{e1}\+\mathrm{e3}$

$Y≔\mathrm{evalDG}\left(\mathrm{e1}-\mathrm{e2}+\mathrm{e3}\right)$

$Y:=\mathrm{e1}-\mathrm{e2}\+\mathrm{e3}$

$\mathrm{Killing}\left(X\,Y\right)$

$-4$

$K≔\mathrm{Killing}\left(\right)$

$S≔\left[\mathrm{e2}\,\mathrm{e3}\right]\:$

$\mathrm{Killing}\left(S\right)$

$\mathrm{KillingForm}\left(\mathrm{Alg1}\right)$

$-2\mathrm{\θ1}\mathrm{\θ1}-2\mathrm{\θ2}\mathrm{\θ2}-2\mathrm{\θ3}\mathrm{\θ3}$